Near-rings : the theory and its applications, Volume 23 (North-Holland Mathematics Studies)

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NEAR-RINGS

This Page Intentionally Left Blank

NORTH-HOLLAND MATHEMATICS STUDIES

Near-hgs The Theory and its Applications GUNTERPILZ lnstitut fur Mathematik Johannes-Kepler- Universitat L inz Linz, Austria

Revised edition

NORTH-HOLLAND PUBLlSHlNG COMPANY AMSTERDAM. NEW YORK.OXFORD

23

Q

North-Holland Publishing Company. 1983

All rights reserved. N o part of this publication may he reproduced, srored in a retrieval system, or rransmitred, in any form or by any means. eleclronic. mechanical, photocopying, recording or otherwise, wirhour the prlorpermission of the copyright owner.

ISBN: 0 7204 0.566 I

First printing 1977 Revised edition 198.3

Piihhshers . NORTH-HOLLAND PUBLISHING COMPANY AMSTERDAM NEW YORK . OXFORD

Sole distrrbutorc for rhe U S A rind Canado ELSEVIER SCIENCE PUBLISHING COMPANY, I N C 52 VANDERBILT AVENUE. NEW YORK. N Y 10017

PRINTED IN THE NETHERLANDS

TO MY BELOVED WIFE GERTl

vi

INTERDEPENDENCE G U I D E T h e n u m b e r s i n d i c a t e t h e o n e s o f t h e p a r a o r a p h e s ; 7a is 5 7 , s e c t i o n a), a n d s o o n . F u l l l i n e s m e a n h e a v y , d o t t e d l i n e s s l i g h t dependencies.(59j is a m e r e c o l l e c t i o n o f results.)

vii

PREFACE TO THE SECOND EDITION

Since t h e appeahence 0 6 t h e 6 h t edition 0 6 t k i b book, a bubbtantiae nwnbm 06 papeu and hedlLet6 on neah-hingb came o u t and new p w ~ 2 0 6 t h e theohy wme bohn. Hence 1 wh vehy pleased when Nohth-Holland o66med me t h e p o b b i b U y t o phoduce an updated, h e v h e d , comeoted and extended vetldion 06 tk ib book on n m - h i n g b , which iA b-tiee t h e o d y one i n t k i b d i d d ( b u t t~ exceUent o t h m t e x ~ %me i n pheparration) At that h e , I ' d i d n ' t know t h e ammount 0 6 wohk I had accepted. T k i s edition contains a fiemendoub numbm 0 6 minoh additions and comections. A@m pm6o~m.h~thede changed, 1 h e a l l y know t h e b i g di,f6mence behueen "countable i n ~ i n i t e "and "uncountable binite" now. In d a d , mobt 0 6 t h e hedURtd edition ahe i n dome way incohpolrated o h at least didcovmed a6tm t h e touched i n tkid edition. A h a , {OWL mote c h a p t w wme added. They c o n c m hegutm nearr-hingb, h e neah-hingb , b i c e w z m neah-hingb and t h e connections b&en neah-hingb and automata. The chapten on polynomial neah-hingb wh bubbtantiaeey eneahged. The .fAt 0 6 n e a h - h g b 06 mall ohdm u ~ bextended by adding b.iY~uOtuhaeindotm d o n s and bome neah-hingb on t h e non-cyclic abeLian ghDupb 06 ohdm b and on Ad. A h a , t k i s edition contains 2 2 2 hemahkable (counteh)examples 0 6 n m h g b . Mobt extensions 0 6 tkis edition me ~aiheuwoven i n t o t h e t e x t ; hence 7 do hope that t k i d hecond edition iA not j u b t a b p U extension 06 t h e 6 h . t editio n The " h e L i g i o u h wah" , i d h i g h t oh t e 6 t n e a h - h g b ahe "b&m", iA 6 a unb a W e d . 1 do t h n k thlLt h i g h t nem-hingb me h i g h t , b u t t h e book en& uLith a con&atoh.q chaptm ubing le6t neah-hingb. Many t h a n k go t o NofLth-Hotland doh the.& o66m and t h e m o d t pleasant coopeh~ak, oLLt .tion. Many coUeagueb conthibuted by m a k i n g .impohtant a e n ~ ~ pointing e h h o u and heading p a h t d 0 6 t h e new manuhchipt. Majoh conhibutionh came ( i n d p h a b d c ohdet) &om G. Behch (Tubingen, Gmmany), J.U. P.M&&um (Edinbwrgh, S c o f h n d ) , S.V.ScoM (Auckland, New Z e d a n d ) , Y.S.So (Taichung, Taiwan) and H.J.Wain& (Clauhthae, G m a n y ) . A b o , many thank go t o G.KoUm and A. Kutzlm dolr t h e i h Q.xc&.&~~ typing j o b . They incohpolrated t h e additions 6 0 bhil@k%j i n t o t h e t e x t t h a t nobody,who only w a d t o head thebe a d d i t i o n s , can didcovm them. L a s t , b u t not at l e a s t , I deeply thank my wide G W . She hdped me mobt.

.

.


ix

FROM THE PREFACE TO THE FtRST EMTiON a "mhg ahe g e n w z e d hingd. Roughly dpoken, a nem-hing ( N , + , . I , &ehe + i.6 not nece.6bafdi.g a b d i a n and wikh o n t y one dinMblLtive

Nm-hhgd

bw,,.

Neah-hingd ahide i n a natuhae way: &tabe t h e be2 Mlr) 0 6 att M p p h ~ g d 06 a gmup (r,+l i n t o &e..t.ed, de6ine additian + point-whely and o a6 cornpabLthn. Then (M(r1 ,+, 0 ) i.6 a n m - h i n g . Even i 6 r h a b e l h n , o n l y one disthibutive &UI iA aeUayb 6utd.ieeed: ( i + g ) q h= 6oh+goh holds by t h e de6inhXon 06 6+g white doh 6o(g+h)= dog+deh we mutd have to adbwne t h a t 4 i.6 a homomohphibm. Anothm cxrunpte i.6 d U p p ! i d by t h e polynonciaed w . 8 . t . addLtion and hub6.titLLtion. A WeU-known tlebUet i n hing theotly dayb that e v a y hing can be embedded into t h e hing Elr) 06 a l l endomotlpL.bm 06 borne a b U n gmup r. Foh n m h g b we pmve ( 1.161 that e v a y nem-&g can be embedded into M(r) 60% dame gmup r. Hmce one migkt v i w hing theohy ab t h e " f i n m t h w h y 04 gmup mappingb", while n m - h i n g b pmvide t h e "non-.&npAIL thwty". SUILpILi6ingly, a t o t 06 ".&nm tle6uLtb" can be L t a n b 6 r n e d ~ t ot h e gmnehae c a e aiteh dLLitabte changes. Foh i n b m c e , t h e "aXomb1' 06 hing theohy, t h e phinritive h i n g d , m e d a d b e d by t h e ~auwusd e n b d y theohem 06 N . 3acobbon (doh h g s w i t h minimUn condition: W e d d m b m - M h t h w h m on bimpte h i n g d ) . foh nean-hiftgb bAni.kb tlebbcLetb cancaning phimit.iue neat-hingd wme 0 M a i n e d v i a t h e u m k 06 6QvQAd aLLthohd ( b u t t h e pmo$b aht to&d..ly d L 6 6 a ~ n t )t:h e hole a$ ffo%(V,Y) doh aingb iA played by M(r) oh dome hekzted t y p e s i n the n m - h h g m e (4.52, 4.54).

t d i s t O h i d y , t h e 6 h t 6tep tolllaha neah-hingb wab an a c O d C h e d Q W h done by V i c h o n i n 1905. ffe dhowed tfmt thehe do ex#t ' ' ~ i c t d bw i t h onty one d i s ~ % M v e W ' ( = n m - 6 i d d 6 ) . Some ylatch thebe nearr-dieeds & h o d up again and phDued t o be uae6LLe i n c o o k d i d z i n g cehtain impohtant daAbeA 06 g w m a c p . h e d ( h e m that lk6cahte'b method 06 coohdinatizing t h e " u b d " pkkne by t h e d i d d 06 heal numbm wa6 one 0 6 t h e m o d t ducceb6~u.t d t e p b .in geomeZq). It wa6 Zabbenhaus who wa6 abte t o d & W e all ~i.n..Lte n m - 6 i e C d ~ ( 1 . 3 4 ) . ~ U m d a y b ,n m - 4 i e l d b ahe a nrigkty tool h~chahactdz&g doubly thanbLtive gtloupd ( 8 . 4 4 1 , inincidence ghoups (8.6%) and Fmb&us gmupb ( 8 . 8 1 ) . Since t h e dwn 06 A3uo endomohppkibm 06 a n o n - a b m gmup ( r , + )i.6 not an endomotlpki6m i n genehat, t h e d e t d E ( r ) 06 aU 6 W e 4wn6 and di6?4menceb 06 endomohphh6 0 6 r m e consklehed. hebpect t o addition

Preface

X

and compobLtion, t h u e E ( r ) '6 ahe neah-hingb belonging t o t h e C h b 06 t h e "dib.thibU.t.iw&y gwnehated" neah-hingb Many p#u2 0 6 t h e we,U-ehtnblhhed fheohy 06 hingb wehe tRand5eNred t o n m hGzgd and n w neat-hing-bpecidic ~ e a t w uw4he dincovetled, bLLieding up a theoay 0 6 nu&-hingb b t e p b y b t e p . Up t o now, about 550 p a p m on Wah-hingb (and n u - d i d d s ) w i t h about 8000 pageb appeahed i n phint, b u t t h e m exha2 no book on t k i s b u b j e o t . Tkid book .thieb t o unidy t h e theohy and L t h feminology and t o g i v e a bgbtematic and w&-asbohted account 06 t h e phuent n t a t e 06 t h e t h e o q . Some aemahn am .to be made: ( a ) GenmaUy, 7 avoided t o g i v e ph006b doh t h e o h m which ahe &hm not &ng t h e main b . & m 06 dibcudb-ion oh a h long ~ ones which contain bpeciae mcthoh seemingly a,oflicable o d y i n .thLh context, cannot be b h p t 4 i e d b y pewhiow hunuets and invo.P,ve many o t h m ( e . g . geornethicat) dC?.tdied, b u t ahe heaiLiey accubib& in t h e f.Ltaatwre. ( b l Sevmak? hulLetd 6oUowing dhom uvLiwm& d g e b m oh &om t h e t h e o h y 0 4 ghoup with mubkple opnehatou ahe c i t e d , bu.t not p v e d i n a d m to dev,i.de hes1Letb which ahe bp&fiic doh n m - h i n g b and Rhobe which ahe

.

not. @L m y 6~wmbeing a m a e coUecLion 0 5 - t & w i d hesuets concaning dome "pathologiCae" b y b t m tU.ithout any application t o o t h m bzmches 06 mathematicn. A p a h t dotun t h e appfic&onb concaning axiomaticb and geomethy mentioned above, bpeciae & b b u 06 d i n i t e n u - h i n g b ( t h e AinLte "plans" nea-hingb) g i v e nw and highey eddi&nt C h b u 06 balanced p a h a w e A m ( 8 . 1 1 7 - 8 . 1 2 4 ) . Moheovm, incomplete block debigm aeheady [uith b& these p l m neah-hingb can be uaed t o chaaactmize Fhobg m u p , hence a h o 6.inite ghouph w i t h d i w e d - p o i n t - d u e a&tomohpkism ghoupb ( 8 . 9 6 , 8.971. 16 r i d a &inite, inwaAhaXy a h p l e n0n-abcL.h gmup, E ( r ) h "phimctive" and t h ee 6 o h e ewmy belb-map 06 r &ixing z m h .thk " b u m 0 6 endomohpkism" [exact @munLLeation i n 7 . 4 7 ) . Anotha v m i o n 06 t h e d e n b a y theohem 4.52 b h o u t h a t t h e d m a y p m p e h t y h ( i n t h e n e a - h i n g - c a e ) bometking Like an i n t m p o l a t i o n phopehty,giwing t h e hesuet that i d a neah-king N (with nome addition& p k o p u d i u l 06 rmppingb on a ghoup r "intehpokk..tes" at zmo and at &a o t h m p k c u t h e n N "intmpakk..tudl &eady at a k b & w u j ( ~ . i n i t & q )many p o d ( 4 . 6 5 ) . m o , n u - h i n g b n i g k t be t h e appkophiate t o o l t o d e w h p a "non-ab&i.an homobgiCae algebha" ( 9 . 2 6 4 ) and dhow up again i n &geb&c topology ( 9 . 2 6 2 1 , 6unctiovd a d y b h ( 9 . 2 6 1 ) and i n

Nem-hing Zheohy h

Preface

xi

categohies xkth ghoup objeotn ( 9 . 2 6 5 ) . FinaUy, t h e a u t b h hopa that neathingo and " n m - k i n g m o d d a b t ( = N-ghOup6) Wiee pmue ;to be u&zdd doh Q n m b m 06 theohiG which .thy t o genehdeize "fineat" hedu&% t o t h e "YPOMf i n e a t cahe", doh . i n h t ~ n ~i en t h e t h e o k i a 06 automata and dynamicat bybtemb (bee § 9 ill, t o make t h e phovehb "16 you Xhy t o non-fineahize, you luiee dind t h e neah-hingb nice" come thue. Fmm t h e h i n g - t h e o h d c d point 06 view, many b i z m e bitua,t.bnts m i h e i n neah-hingb. f o h example, not evehy Le6.t i d e d .ih a hubneah-hing. Howeveh, thehe me bevehd bpLLcLte appUcaChm 06 nm-hingn 2O hi4 t h e o w (doh t h e fim-Rhtg-hehuk% dhow what h con&ined 2O hingn and whdt , .ih n o t ) and t o d v m d dgebha (becawe a h i g h pacentage 06 dedinitiom and ha& 06 t h e neah-hing theohy c a y oveh t o u n b ~ ddg e b h a ) . On quotations: Redehenced t o o t h m b e o t i o n h ahe done e.g. b y " Z d ) " meaning " 5 2 , beckLon d ) " oh by " 2 d 3 ) " abbhevhting "52, section d ) , numbeh 3 ) " , Numbm 6ok'lowing nama od authom hedeh t o t h e bibi!.iogmphy at t h e end 06 given, a.U p a p e a 06 t k i b authoh t h e book. 16 o n l y t h e authoh'b name c i t e d i n t h e bib&ogmphy ahe meant. T k i e bibfiagtraphy a b d d be ~ a i h a y cornplWe ah A a h ah nm-hingb ahe concehned. F O h nem-dieL& and hdaZed bubjectb we o n l y U t .thohe papeh?, which dihec;teY in@uenced t h e mat&L i n t k i b book. T k i n bibfioghaphy ~ a compiled 6 i n domm yea^ by J.C&y, G.B&ch, J.Mdone, H.UeatheMy and t h e authoh. Nama i n b u c k & hedeh t o t h e l?,ibt 06 "Supp&nwUa~y WOht2A11which c o n t a - h t h e non-neaA-h&g-papm oited tkio book. Sevehd hebu&h i n t k i n monaghaph ahe n w oh i n a new (and hope&Uy hnphoved) &hm UlithoLLt b p e c i d noLice. In t h e beginning 06 p o o 6 b thehe h no hepetition 06 t h e ahbump-tiom ( t o have space). l I * I t and l t e f I mean t h a t t h e diheotion indicated .ih h e a t e d at moment

(in p o o i b 06 equivdenced).

It 0 a pLmwle t o thank Utr. E . Fhedh.ibbon 06 t h e ed.Ltohid btadd 06 t h e Nohth-HuLlund Pubuhing Company and t h e heviweh doh a pLeaant cuopehhation and a L o t oh uhadul buggebXionh. Many thank4 go &a t o h. Hobpodah doh

xi i

Preface

he& excelLent typing j o b and t o G . ReLbch, Y.-S.

So, H.E. ReRC, J.D.P.

Meldhum and t o M . L . Holcombe d o h heading p a & 06 t h e manwchipt and ptouiding wed& kid and &po.ILtant commenL5. h(o4.t 06 CLee 1 haue t o thank my wide doh h a pcLtience and endutance i n fiuing w d h an aboevLt-mindedhunbund i n t h e p a t ye&. And now good Puck and much dun w L t h nem-hingb!

Remahh: A "Nedt~-h.CngNe~14leeMeh" c o m a ouA once oh twice a yeat, c o n t a i n i n g in@~ma-tiona b o u t t h e hecent deuelvpmed i n t h e theo.ty 04 n e m hingb. 16 you want t v o b t a i n c o p i e ~ ,w h i t e t o A. Obwaed o h t o t h e aLLthoh 06 tkin book.

xi ii

CONTENTS

..................................... ............................. ..................... .........................................

Interdependence guide Preface t o t h e second e d i t i o n From t h e p r e f a c e t o t h e f i r s t e d i t i o n

5 0 PREREQUISITES . PART I :

4

NEAR-RINGS

................... ............. ...................................... ........................................ ................................... ........... .................................... ............................... ...................................... .... ......................... .................... ......................................... ............................... ....................................... ...................... ........................... .............................. I D E A L THEORY .......................................... a ) Sums ............................................... 1 ) Sums a n d d i r e c t sums ............................ 2 ) D i s t r i b u t i v e sums ............................... b ) C h a i n c o n d i t i o n s ................................... c ) D e c o m p o s i t i o n t h e o r e m s ............................. d ) P r i m e i d e a l s ....................................... 1 ) P r o d u c t s o f s u b s e t s ............................. 2 ) P r i m e i d e a l s .................................... 3 ) S e m i p r i m e i d e a l s ................................ e ) N i l a n d n i l p o t e n t .................................. a ) Fundamental d e f i n i t i o n s and p r o p e r t i e s 1 ) Near-rings 2 ) N-groups 3) Substructures 4 ) Homomorphisms and i d e a l - l i k e c o n c e p t s 5) Annihilators 6 ) Generated o b j e c t s b ) Constructions 1 ) P r o d u c t s . d i r e c t sums a n d s u b d i r e c t p r o d u c t s 2 ) Near-rings o f quotients 3 ) F r e e n e a r - r i n g s and N - g r o u p s c ) Embeddings 1 ) Embedding i n M(T) 2 ) More beds d ) Some a x i o m a t i c c o n s i d e r a t i o n s 1 ) Miscellaneous r e s u l t s 2 ) Related structures

. P A R T I1 :

4

1

FOR B___EGINNERS

I T H E E L E M E N T A R Y THEORY OF N E A R - R I N G S

h 2

vi vii i x

6 7 7 13 14 15 20 23 24 24 26 29 33 33 37 38 38 41 43 44 44 49 50

53 61 61 62 66

69

STRUCTURE T H E O-~ RY

...................... T y p e s o f N - g r o u p s .................................. C h a n g e o f t h e n e a r - r i n g ............................ M o d u l a r i t y ......................................... Q u a s i r e g u l a r i t y .................................... I d e m p o t e n t s ........................................

3 ELEMENTS OF .

THE STRUCTURE THEORY

a) b) c) d) e) f ) More o n m i n i m a l i t y

.................................

74

75 81 84 89 91 95

xiv

Contents

................................. a ) G e n e r a l ........................................... 1 ) Definitions a n d elementary results . . . . . . . . . . . . . 2 ) T h e centralizer ................................ 3 ) I n d e p e n d e n c e a n d density ....................... b ) 0-primitive n e a r - r i n g s ............................ c ) I-primitive near-rings ............................ d ) 2-primitive near-rings ............................ 1 ) 2-primitive n e a r - r i n g s ......................... 2 ) 2-primitive near-rings with identity . . . . . . . . . . . 3 ) 2-primitive zero-symmetric n e a r - r i n g s with identity a n d a m i n i m a l l e f t i d e a l .............. 4 ) 2-primitive n e a r - r i n g s with identity a n d m i n i m u m condition .............................. 5 ) A n application to interpolation theory . . . . . . . . . R A D I C A L T H E O R Y .......................................

6 4 P R I M I T I V E NEAR-RINGS

6 5

..................... ...... ...

102 103 103 106 110 115 120 124 124 126 130 131 133

135

.....................................

Jacobson-type 1 ) Definitions a n d characterizations o f the r a d i c a l s 1 3 6 near-rings ....................

.................................................... Jacobson-type ..................................... go 21. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . J 1 ............................................... ................................................... ..................................................... .. ............ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ......................................................

P A R T 111:

0

S P E C I A L CLASSES O F N E A R - R I N G S

.................. Elementary ........................................ Some axiomatics ................................... Constructions o f d . g . n e a r - r i n g s . . . . . . . . . . . . . . . . . . Distributively generated n e a r - r i n g s with finiteness conditions ........................................ " F r e e " distributively generated near-rings . . . . . . . . 0 - g r o u p s a n d ( N . D ) . g r o u p s .........................

6 DISTRIBUTIVELY GENERATED NEAR-RINGS

a) b)

c) d)

e)

f ) 9 ) Structure theory

0

7

.................................. T R A N S F O R M A T I O N N E A R - R I N G S ............................ a ) M:(r) ............................................. b ) M(r) a n d Mo(r) ....................................

............................... ............................. functions ........... ........................................... ........................................... ........................... ...........................................

E(r). A(r) and I ( r ) d ) P o l y n o m i a l near-rings 1 ) Polynomials a n d polynomial 2 ) R[xl 3) P(R) 4 ) I d e a l theory i n R [ x l 5) F[xl c)

170

171 174 176 178 180 182 184

188 189 197 206 215 215 218 219 220 223

Contents

6 ) r[xl and

P ( r ) ...................................

7 ) P o l y n o m i a l s o v e r i ; - g r o u p s ....................... 8) C o n c l u d i n g r e m a r k s ..............................

5 8

5

9

.

230 233 244

..................... 248 a ) N e a r - f i e l d s ........................................ 249 1 ) C o n d i t i o n s t o be a n e a r - f i e l d . . . . . . . . . . . . . . . . . . . 2 4 9 2 ) T h e a d d i t i v e group of a n e a r - f i e l d .............. 2 5 1 3 ) The c e n t e r a n d t h e k e r n e l o f a n e a r - f i e l d ....... 2 5 3 4 ) D i c k s o n n e a r - f i e l d s ............................. 254 5 ) N e a r - f i e l d s a n d d o u b l y t r a n s i t i v e g r o u p s . . . . . . . . 258 6 ) Normal n e a r - f i e l d s a n d i n c i d e n c e g r o u p s . . . . . . . . . 260 7 ) P l a n a r n e a r - f i e l d s .............................. 265 b ) P l a n a r n e a r - r i n g s .................................. 268 1 ) T h e s t r u c t u r e o f p l a n a r n e a r - r i n g s . . . . . . . . . . . . . . 268 2 ) P l a n a r n e a r - r i n g s a n d BIB-designs ............... 276 M O R E CLASSES OF N E A R - R I N G S ............................ 287 a ) I F P - n e a r - r i n g s ..................................... 288 1 ) I F P - n e a r - r i n g s .................................. 288 2 ) p - n e a r - r i n g s .................................... 298 3 ) B o o l e a n n e a r - r i n g s .............................. 300 b ) N e a r - r i n g s w i t h o u t ................................. 301 1 ) N e a r - r i n g s w i t h o u t n i l p o t e n t e l e m e n t s . . . . . . . . . . . 301 2 ) N e a r - r i n g s w i t h o u t z e r o d i v i s o r s ................ 305 313 c ) A f f i n e n e a r - r i n g s .................................. d ) N e a r - r i n g s on g i v e n g r o u p s ......................... 321 1 ) M u l t i p l i c a t i o n s o n a g r o u p ...................... 321 2 ) N e a r - r i n g s o n s i i i i p l e a n d o n c y c l i c g r o u p s . . . . . . . 325 3 ) N e a r - r i n g s w i t h i d e n t i t i e s o n g i v e n g r o u p s . . . . . . 327 4 ) N e a r - r i n g s w i t h o t h e r p r o p e r t i e s o n g i v e n g r o u p s . 330 333 e ) O r d e r e d n e a r - r i n g s ................................. 345 f ) R e g u l a r n e a r - r i n g s ................................. g ) Tame n e a r - r i n g s .................................... 350 h ) B i c e n t r a l i z e r n e a r - r i n g s ........................... 361 i ) N e a r - r i n g s a n d a u t o m a t a ............................ 378 j) M i s c e l l a n e o u s ...................................... 392 N E A R - F I E L D S A N D P L A N A R NEAR-RINGS

APPENDIX

.

xv

..................................................

N e a r - r i n g s o f low o r d e r ............................... 2 2 2 r e m a r k a b l e e x a m p l e s and c o u n t e r e x a m p l e s . . . . . . . . . . . L i s t o f some open p r o b l e m s ............................ B i b l i o g r a p h y .......................................... S u p p l e m e n t a r y works ................................... L i s t o f s y m b o l s and a b b r e v i a t i o n s ..................... I n d e x .................................................

404 405 426 435 437 464 465 467


1

80

PREREQUISITES

For t h e c o n c e p t of s e t s we can use any one of t h e usual s e t t h e o r i e s w i t h t h e axiom o f c h o i c e and u s i n g c l a s s e s . I n o r d e r t o a v o i d l o g i c a l d i f f i c u l t i e s a s much a s p o s s i b l e , we u s e s t a t e m e n t s a b o u t c l a s s e s only a s a b b r e v i a t i o n s of " l e s s obscure o n e s " . F o r i n s t a n c e , i f 7 d e n o t e s t h e c l a s s of a l l f i n i t e s e t s , "FE$" i s only a n a b b r e v i a t i o n f o r "F i s a f i n i t e s e t " . " ~ x E A " stands f o r " t h e r e e x i s t s a n X E A " , "3X E A " f o r " t h e r e e x i s t s e x a c t l y one X E A " and " t / x ~ A " f o r " f o r a l l X E A " .

I n c l u s i o n w i l l be d e n o t e d by c , s t r i c t i n c l u s i o n by =. 0 w i l l d e n o t e t h e ( a n ) empty s e t a n d ilA t h e power s e t o f A ; i f A i ( i E I ) i s a c o l l e c t i o n o f s e t s , we w i l l w r i t e t h e e l e m e n t s o f Ai:=A X A i a s ( . . . , a i , . . . ) , where a i € A i . I f a l l A i 5 A t h e n i€ 1 i EB and a l s o fl A i : = A . A \ B i s t h e s e t - t h e o r e t i c d i f f e r e n c e . I f i €0 % i s a n equivalence r e l a t i o n on the s e t A , A / % w i l l be t h e f a c t o r s e t of A w . r . t . 2, a n d n:A -* A/?. w i l l be the canonical projection.

x

The s e t s o f a l l n a t u r a l numbers w i l l be d e n o t e d by I N , t h e n a t u r a l numbers t o g e t h e r w i t h 0 by INo , t h e prime numbers by IP, t h e i n t e g e r s by Z, t h e r a t i o n a l s by Q , t h e r e a l s by lR a n d t h e complex numbers by b .

I f f i s a f u n c t i o n from A t o B a n d i f A l c A

then

f/A1 be t h e r e s t r i c t i o n o f f t o A1 a n d f ( A 1 ) w i l l d e n o t e t h e imaqe o f A 1 under f . B A w i l l be t h e s e t o f a l l maps from A t o B . I f BSA, i : B + A w i l l be r e s e r v e d f o r t h e i n c l u s i o n map. I f A i s any s e t c o n t a i n i n q something l i k e a " z e r o e l e m e n t " 0 , lN, w h i l e P * i s n o t d e f i n e d . A * w i l l d e n o t e A \ { O ) . S o e . g . =:"

$ 0 PREREQUISITES

2

" F i e l d " w i l l a l w a y s mean " s k e w - f i e l d " . The s y r f l n i e t r i c ( a l t e r n a t i n g ) g r o u p on n s y m b o l s w i l l be d e n o t e d by S n ( A n , r e s p e c t i v e l y ) . The i n t e g e r s modulo n w i l l be w r i t t e n a s 1, a n d r e p r e s e n n-1). t e d by Zn = { O y l ¶ . . . ,

We need an a b s t r a c t v e r s i o n o f " g e n e r a t e d o b j e c t s " :

A52A

0.1 DEFINITION i s c a l l e d a Moore-system (Dubreil-Dub r e i l - J a c o t i n ) on A i f (a) A E ~ . (b) i s closed w.r.t. a r b i t r a r y i n t e r s e c t i o n s . 0.2 PROPOSITION If [B]&

:=

fl M MEM M 'B

&

i s a M o o r e - s y s t e m on A a n d i f BGA t h e n i s t h e s m a l l e s t e l e m e n t o f /CC(w.r.t. 5 ) c o n -

t a i n i n g 6.

0 . 3 DEFINITION L e t t h e n o t a t i o n be a s a b o v e . ( a ) [ElA i s c a l l e d t h e element o f A which i s a e n e r a t e d

by 6.. i s called f i n i t e l y oenerable (f.q.) f i n i t e subset B o f A w i t h [B]& = A.

( b ) AE,~!

i f there i s a

A

0.4 D E F I N I T I O N A Moore-system i s called inductive i f H c o n t a i n s t h e union of e v e r y c h a i n of elements o f A .

0 . 5 EXAMPLES ( a ) Z A i s a n i n d u c t i v e Y o o r e - s y s t e m on A . ( b ) The s e t o f a l l s u b q r o u p s o f a q r o u p r i s a n i n d u c t i v e M o o r e - s y s t e m on r . ( c ) The s e t o f a l l c l o s e d s u b s e t s o f a t o p o l o o i c a l s p a c e T i s a M o o r e - s y s t e m on T w h i c h i s n o t i n d u c t i v e i n q e n e ral. We now turn t o c h a i n c o n d i t i o n s .

3

$ 0 PREREQUISITES

0.6 DEFINITION ----fulfill

A ( p a r t i a l l y ) ordered set

(A,

al

...

o f elements o f A terminates

a f t e r f i n i t e l y many s t e p s 2 a2 2

al

chain

...

3

(or,

nEIN :

equivalently, a n = an+l

=

f o r each

...

).

0 . 8 DEFINITION L i n e a r l y o r d e r e d s e t s w i t h t h e minimum c o n d i t i o n a r e c a l l e d we1 1 - o r d e r e d .

3.9 R E M A R K I n r e p l a c i n g

,

by 2

5

"maximum c o n d i t i o n " ,

we g e t t h e c o n c e p t s o f

"ascendins chain covdi t i o n "

( A.- C C )

a n d " i n v e r s e we1 1 - o r d e r " . E v e r y non-empty s u b s e t o f an o r d e r e d s e t w i t h t h e minimum ( m a x i m u m ) c o n d i t i o n h a s t h e same p r o p e r t y . be a Moore-system o n t h e s e t A .

0 . 1 0 PROPOSITION L e t

( h ! , ~f u) l f i l l s Ifk

t h e A C C =>

i s inductive,

Proof. Let

(4,~ h a v)e

i s n o t f.g. trary cess,

[{bl}lA

B2:=

=:B1

Now l e t

/cI

ment M o f

Ml=M2=M3=

A,

[{bl,b2}]Ac

BcA.

ME&

T a k e some a r b i -

B1=B2cB3=

...

of

a contradiction..

be i n d u c t i v e and suppose t h a t e v e r y e l e -

A ...

i s f.g..

U Mi E A b e g e n e r a t e d b y ( s a y ) icLN B u t t h e r e i s some kEIN w i t h t h e p r o p e r -

A?. L e t M:=

{al,

...,a n ) . {al,

Assume m o r e o v e r t h a t

i s a s t r i c t i n f i n i t e chain o f elements

of

ty t h a t

i s f.g..

$. M . T a k e some b2El'r\B1 M. C o n t i n u i n g t h i s p r o -

one g e t s an i n f i n i t e c h a i n

elements o f

A

t h e A C C a n d a s s u m e t h a t some

and g e n e r a t e d b y

blEB.

and f o r m

every element o f

t h e converse also holds.

...,a n I C M k ,

again a contradiction.

s o we g e t

Mk = M, w h i c h i s

4

80 PREREQUISITES

F i n a l l y , i t s h o u l d be remarked t h a t i n g e n e r a l we use small l e t t e r s f o r e l e m e n t s , c a p i t a l s f o r s e t s and s c r i p t l e t t e r s f o r c o l l e c t i o n s of s e t s .

PART I NEAR-RINGS FOR BEGINNERS

31

THE ELEMENTARY THEORY OF NEAR-RINGS

3 2 IDEAL THEORY

6

§ 1 THE ELEMENTARY THEORY OF NEAR-RINGS

a ) F U N D A M E N T A L DEFINITIONS A N D PROPERTIES

N e a r - r i n g s a r e g e n e r a l i z e d r i n q s : a d d i t i o n n e e d s n o t be comm u t a t i v e and ( m o r e i m p o r t a n t ) o n l y o n e d i s t r i b u t i v e l a w i s p o s t u l ated. Examples o f n e a r - r i n g s a r e (a) the set M(T) o f a l l m a p p i n g s on a n ( a d d i t i v e l y w r i t t e n ) g r o u p r w i t h p o i n t w i s e a d d i t i o n and c o m p o s i t i o n ; ( b ) t h e polynomials R[X] ( R a commutative r i n g w i t h i d e n t i t y ) under a d d i t i o n a n d s u b s t i t u t i o n ; ( c ) an a r b i t r a r y a d d i t i v e l y w r i t t e n q r o u p w i t h z e r o m u l t i p l i c a tion; its w e l l a s many o t h e r s . S i m i l a r t o r i n g t h e o r y , " m o d u l e s o v e r a n e a r - r i n q PI" ( ' ' n e a r m o d u l e s " or ' I N - g r o u p s " ) w i l l be i n t r o d u c e d . They p l a y an i m portant r 6 l e in t h e theory o f near-rings. This s e c t i o n contains t h e b a s i c d e f i n i t i o n s , examples a n d p r o p e r t i e s o f n e a r - r i n q s and N - g r o u p s , a n d o f s u b s t r u c t u r e s a n d i d e a l - l i k e o b j e c t s i n these kinds o f a l q e b r a s . S i n c e n e a r - r i n g s a n d Pi-groups ( w i t h a z e r o - s y m m e t r i c N ) a r e s p e c i a l c l a s s e s o f R - q r o u p s ( q r o u p s w i t h mu1 t i p l e o p e r a t o r s ) , a w h o l e bunch o f c o n c e p t s a n d r e s u l t s i s " a p r i o r i " a v a i l a b l e . Compared w i t h r i n g t h e o r y , some c o m p l i c a t i o n s a r i s e : a n e l e m e n t m u l t i p l i e d by 0 i s n o t 0 i n g e n e r a l , t h e c h a r a c t e r i z a t i o n o f i d e a l s i s a l i t t l e b i t more c o m p l i c a t e d , i d e a l s a r e n o t a l w a y s s u b a l g e b r a s , and so o n .

7

l a Fundamental definitions and properties

1.)

N E A R - R I NGS

1.1 DEFINITION A n e a r - r i n o i s a s e t N t o q e t h e r w i t h t w o b i n a r y operations

and

"+I'

such t h a t

'I.''

(a)

(N,+)

i s a group (not necessarily abelian)

(b)

(N,.)

i s a semiqroup

(c)

( n + n ) . n 3 = nl.n3+n2.n3 1 2 tributive law").

1.2 REMARKS I n v i e w o f ( c ) , " r i g h t near-ri no". (cl)

("riqht dis-

n1,n2,n3€N:

V

Postulating

+ n ) = nl.n2+nl.n3 2 3 one g e t s " l e f t n e a r - r i n n s " .

n1,n2,n3cN:

instead o f (c),

one speaks more p r e c i s e l y o f a

nl.(n

The t h e o r y

runs completely p a r a l l e l i n both cases, o f course;

so one

can d e c i d e t o use j u s t one v e r s i o n . Although l e f t n e a r - r i n q s a r e more f r e q u e n t l y used i n t h e l i t e r a t u r e u p t o now,

we w i l l u s e r i g h t n e a r - r i n a s :

* T h e l e f t d i s t r i b u t i v e l a w i s i n some way u n n a t u r a l i n n e a r - r i n g s o f f u n c t i o n s ( t h e most i m p o r t a n t examples) and e s p e c i a l l y unmotivated i n n e a r - r i n q s o f p o l y n o m i a l s and formal

power s e r i e s .

*An a d - h o c - t e s t

d o n e b y t h e a u t h o r showed

80% o f t h e books i n which rinq-modules r 6 l e use left-modules,

t h a t about

p l a y an important

w h i c h a r e a l s o more f a m i l i a r f r o m

t h e t h e o r y o f v e c t o r spaces.

I n 1.18,

we w i l l s e e t h a t

c h o o s i n g l e f t N-groups f o r c e s one t o use r i q h t n e a r - r i n q s . *The r i g h t d i s t r i b u t i v e law i s e x c l u s i v e l y used i n papers

on t h e c l o s e l y r e l a t e d c o n c e p t o f c o m p o s i t i o n r i n q s (which were s y s t e m a t i c a l l y s t u d i e d p r i o r t o n e a r - r i n q s ! ) . 1.3

NOTATION N e a r - r i n g s w i l l u s u a l l y b e d e n o t e d b y N , N ' , N 1 s i m i l a r symbols. on w i l l write

We a b b r e v i a t e ( N , + , . )

by

N.

i n most cases be i n d i c a t e d by j u x t a p o s i t i o n ; nln2

instead o f

n1.n2.

or

Multiplicati-

s o we

I n dealinq w i t h general

n e a r - r i n g s t h s n e u t r a l e l e m e n t o f (N,+)

w i l l be denoted

b y O . I N 1 w i l l be t h e o r d e r o f t h e n e a r - r i n g N.


01

8

The t e r m " n e a r - r i n g "

will b e d e n o t e d a near-ring,

by

-

w i l l o f t e n be a b b r e v i a t e d b y " n r . " .

T h r o u g h o u t t h i s monograph,

n. I f

the class o f a l l near-rings

"N" a p p e a r s ,

it

will a l w a y s b e

without further notice.

1 . 4 EXAMPLES (a) Let

r

be an a d d i t i v e l y w r i t t e n ( b u t n o t n e c e s s a r i l y

abelian)

group w i t h zero

o ("omykron").

f o l l o w i n g s e t s o f mappings f r o m

into

r

are nr.'s

substitution:

under p o i n t w i s e a d d i t i o n and

M(r):= { f : r + r } =rr . Fio(r):= { f : r + r l f ( o )

r

Then t h e

= 01.

If:r+r(f i s constant].

Mc(r):=

M0c ( r ) : = { f 6 : r + r 1 6 E r A f 6 ( y ) = (Evidently,r , Mc(r) a n d M:(r)

{:

i ff y+o i y=o I . are isomorphic

groups). Mcont(r):=

{f:r+I'lf i s c o n t i n u o u s ]

(r

a topological

group). A n o t h e r r e l a t e d example i s

I

Mdiff(IR):= (f:IR+ IR f i s differentiable), while the r e a l f u n c t i o n s h a v i n g an i n d e f i n i t e i n t e g r a l do n o t form a near-ring (they are n o t closed w . r . t .

composi-

tion). For

SsEnd(r)

MS(r):=

{f:r+rl

define

WS~S:

where 6 These (b) Let

M(r)

=

E v i d e n t l y , MIid1(r)

f o s = sOf1.

and

M{6](r)

= Mo(r),

i s t h e z e r o endomorphism.

M S ( r ) ' s w i l l become v e r y i m p o r t a n t i n 9 4 .

r

be as above.

N e a r - r i n g s on

r

a r e e.g.

(I-,+,*)

with

yi-6 = o

for all

y,6EI';

(r,+,+)

with

y+6 = y

for all

y,6cr.

M o r e g e n e r a l l y , t a k e some s u b s e t A o f r a n d d e f i n e A T h e n (r,+.*,,) i s a n e a r - r i n g i f y'A6:= ii ff 66 E4 A '

{;

04A.

(Multiplications o f t h i s type are called the " t r i v i a l ones" i n Malone ( 3 ) .

because t h e y a r e e x a c t l y t h o s e

ones w h i c h can be d e f i n e d o n any group, group i n t o a near-rinq.)

makinq t h i s

9

l a Fundamental definitions and properties Now l e t G b e a f i x e d - p o i n t - f r e e

r

(i.e.

Choose a n y s u b s e t

{Bi(iE.Il

z e r o o r b i t s o f G on

automorphism group on

g ( y ) = y -=> (y = o v q = i d ) ) .

bgEGbyEr:

r

o f t h e s e t o f a l l non-

(Betsch c a l l e d these o r b i t s

" 1 - o r b i t s " and t h e o t h e r ones " 0 - o r b i t s " ) ; m o r e o v e r , choose any s e t o f r e p r e s e n t a t i v e s { b i ~ B i \ i E I l = : B and d e f i n e

yoB6 t o b e o i f

64

u

Bi

and t o b e

=g6(y)

i E I

i f 6 i s i n some Bi, w h e r e g 6 i s t h e u n i q u e a u t o m o r p h i s m i n G s e n d i n q bi i n t o 6 :

Then

(r,t,=B)

i s a n e a r - r i n g as one sees b y l o o k i n g

a t t h e d i f f e r e n t p o s s i b l e c a s e s . These t y p e s o f n r . ' s were i n t r o d l i c e d by F e r r e r o ( 5 ) and w i l l p r o v e u s e f u l i n t h e t h e o r y o f p l a n a r and i n t e g r a l n e a r - r i n g s . Anyhow,

o n e s e e s t h a t e v e r y g r o u p c a n b e made i n t o a

n e a r - r i n g i n v a r i o u s w a y s . See a l s o O l i v i e r ( 2 ) . ( c ) L e t V b e a v e c t o r s p a c e o v e r some f i e l d F. C a l l a s u s u a l a map V + V a n a f f i n e map i f i t i s t h e s u m o f a l i n e a r and a c o n s t a n t one. The s e t M a f f ( V ) of all I

a f f i n e maps i s a g a i n a n e a r - r i n g ( o p e r a t i o n s a s i n (a)). ( d ) L e t R be a c o m m u t a t i v e r i n g w i t h i d e n t i t y .

are

(R[x].t,O)

and

(R[[~]],t,o),

where

Near-rings O

means

s u b s t i t u t i o n . Another n e a r - r i n g i s formed by t h e s e t P ( R ) o f a l l polynomial

f u n c t i o n s on R w i t h t h e opera-

t i o n s as i n ( a ) ( s e e 5 7 d ) ) .

( e ) O f course, every r i n g i s a near-rinq. 1.5 P R O P O S I T I O N

bn,n'EN:

Proof: as for r i n g s .

On = 0 A ( - n ) n '

= -nn'.

51

10


1 . 6 REMARK A s most o f o u r examples s h o w , n O = 0 and n ( - n l ) = -nn' d o n o t h o l d i n g e n e r a l . F o r i n s t a n c e , i n M(r) f o O = 0 means t h a t " f g o e s t h r o u g h t h e o r i g i n " a n d f o ( - f ' ) = - f o f ' means t h a t ' I f i s a n o d d f u n c t i o n " .

=

I _

One t h e r e f o r e d e f i n e s f o r a n e a r - r i n g N : 1 . 7 DEFINITION rt ( a ) N o : = I n E N l n O = 0 1 i s c a l l e d t h e z e r o - s y m m e t r i c p a-

of N. ( n E N l n O = n ) = (ncNlbl the constant p a r t of N .

( b ) Nc:=

N o and N C a r e i t s e l f n e a r - r i n g s 1.8 EXAMPLES

(M(r))o

= M0(r);

n'EN:

nn' = n)

i s called

(see 1.22 ( a ) ) . (M(I'))c = Mc(r).

1 . 9 DEFINITION N E ? ~ i s c a l l e d z e r o s y m m e t r i c

(constant) i f

N = No

( N = Nc, respectively). stand f o r t h e c l a s s e s of a l l zerosymmetric (constant) near-rings.

11,

(qc)

37,

a r e ( n o t a t i o n as i n 1 . 4 ) M o ( r ) , MS(r) i f ~ E S , every r i n g . M C ( r ) c n c , w h i l e M(r) o r R [ x ] a r e n e i t h e r i n 9, n o r i n 3,. C f . A d l e r ( I ) , p . 6 1 0 .

1.10 E X A M P L E S E l e m e n t s o f

1.11 DEFINITIONS The f o l l o w i n g c o n c e p t s a r e d e f i n e d a s i n r i n g

theory: l e f t ( r i g h t , - ) i d e n t i t i e s , l e f t ( r i g h t , - ) invertible elements, l e f t ( r i g h t , - ) cancellable elements, l e f t ( r i g h t , - ) z e r o d i v i s o r s , idempotent and n i l p o t e n t e l e m e n t s . M o r e o v e r , c a l l dcN d i s t r i b u t i v e i f n,n'EN: d(n+n') = dntdn'. Let N d : = { d E N / d i s d i s t r i butive}. Let 9,be t h e c l a s s o f a l l n e a r - r i n g s w i t h i d e n t i t y ( u s u a l l y d e n o t e d by 1 ) .

v


11

1 . 1 2 EXAMPLES The i d e n t i t y f u n c t i o n s e r v e s a s a n i d e n t i t y i n M(r) a n d Mo(r). I n v e r t i b l e i n t h e s e n e a r - r i n g s a r e e x a c t l y t h e b i j e c t i v e f u n c t i o n s . 2 x i s an example o f a n i l p o t e n t eleinent i n Z4[x]. Cartan ( 1 ) c h a r a c t e L i z e d a l l i n v e r t i b l e elements i n ( F [ [ x ] ! ) o , F a field: 1 aixi i=l h a s an i n v e r s e i n ( F [ [ X ] ] ) ~ ( w . r . t . 0 ) i f f al=/=O. I f N = M a f f ( V ) t h e n N d = HomF(V,V). I f N i s a r i n g t h e n N = N d . I t i s c l e a r t h a t NdcNo. I f N h a s a n i d e n t i t y 1 then l € N 0 . The n e x t a s s e r t i o n s t e m s f r o m B e r m a n - S i l v e r m a n ( 1 ) . G e n e r a l i z a t i o n s c a n be f o u n d i n K a a r l i ( 4 ) , L y o n s (4), M i r o n - S t e f a n e s c u ( I ) , R a m a k o t a i a h - R e d d y ( I ) , Zand ( l ) , ( Z ) . 1 . 1 3 PROPOSITION I f eEN i s i d e m p o t e n t t h e n we g e t J " P e i r c e decomposition": n = x 0+x 1' ncN ~ x o E I : x E N I x e = O ly x l € N e : Taking e = 0 one g e t s n = no+n C tf n E N % " E N O 3 n C € N c : Hence ( N , + ) = ( N o , + ) + ( N c , + ) a n d N o n N c = lo!. P r o o f . n = ( n - n e ) + n e w i l l do t h e d e c o m p o s i t i o n j o b . I f n = x o t x l = x 0' + x i w i t h x o , x ; ~ { x ~ N [ x e = O ) a n d x1 = y l e , = y i e ENe t h e n n e = x e = x ' e . B u t 1 1 x l e = y 1e e = y e = x 1 a n d x i e = I t follows t h a t x1 = a n d x 0 = x;.

.

xi

xi

xi.

1 . 1 4 DEFINITIONS Let N be a n e a r - r i n g . I f ( N , + ) i s a b e l i a n we c a l l N an a b e l i a n n e a r - r i n a ; i f ( N , . ) i s c o m m u t a t i v e we c a l l N i t s e l f a c o m m u t a t i v e n e a r r i n g . I f N = N d , N i s s a i d t o be d i s t r i b u t i v e . I f a l l nonz e r o e l e m e n t s o f N a r e l e f t ( r i g h t , - ) c a n c e l l a b l e , we s a y t h a t N f u l f i l l s the l e f t ( r i q h t , - ) c a n c e l l a t i o n law. N i s i n t e g r a l i f N h a s no n o n - z e r o d i v i s o r s o f z e r o . I f (N*=Ff\{OI,.) i s a g r o u p , N i s c a l l e d a n e a r - f i e l d ( a b b r e v i a t i o n : 6) A n. e a r - r i n g which i s n o t a r i n ? w i l l be r e f e r r e d t o a s a n o n - r i n q . S i m i l a r l y , a n o n - f i e l d i s a n f . w h i c h i s no f i e l d . A n e a r - r i n g w i t h t h e p r o p e r t y t h a t Nd generates (N,+) i s called a d i s t r i b u t i v e l y qenerated near-rinq ( d q n r . ) .

8 1 T H E E L E M E N T A R Y T H E O R Y OF N E A R - R I N G S

12

M(r)

1.15 EXAMPLES ( N o t a t i o n a s i n 1.4) abelian.

(r,+,o)

i s abelian iff

r

i s

s e r v e s as a n example o f a c o m m u t a t i v e

and d i s t r i b u t i v e n o n - r i n g ,

(r,+,*)

while

i s integral.

(r,t,mB) i s i n t e g r a l i f f a l l

I n the language o f 1.4(b),

non-zero o r b i t s are "1-orbits".

(Z2,t)

with

0 - 0 = 0.1

= 0,

1.0 = 1.1 = 1 i s a n f . A l l o t h e r n f ' s a r e z e r o - s y m m e t r i c . L e t r b e a g r o u p . I f r i s n o t a b e l i a n , t h e sum o f t w o endomorphisms i s n o t n e c e s s a r i l y an endomorphism any more. B u t t h e s e t o f a l l ( f i n i t e ) sums a n d d i f f e r q n c e s o f e n d o morphisms of r i s c l o s e d under a d d i t i o n and c o m p o s i t i o n and forms a dgnr.

E(r).

1.16 H I S T O R I C REMARKS N e a r - f i e l d s w e r e t h e f i r s t n r ' s c o n s i d e r e d i n the literature.

I n 1905, Dickson (1),(2)

changed t h e

m u l t i p l i c a t i o n i n a f i e l d i n o r d e r t o g e t examples o f "one-sided d i s t r i b u t i v e f i e l d s "

(= nf's)

showing t h a t t h e

second d i s t r i b u t i v e l a w does n o t f o l l o w f r o m t h e r e m a i n i n g axioms f o r a ( s k e w - ) f i e l d . I'

H i s "changed f i e l d s " a r e c a l l e d

D i c k s o n n f ' s " ( s e e $8 ( a ) 4 )).

A c o u p l e o f y e a r s l a t e r Veblen and Wedderburn

started

t o use n f ' s t o c o o r d i n a t i z e c e r t a i n k i n d s o f geometric planes

.

I n 1936, Zassenhaus (1) d e t e r m i n e d a l l f i n i t e n f ' s : have o r d e r

p n ( P E P , nEIN)

cases) Dickson n f ' s .

they

and a r e ( u p t o 7 e x c e p t i o n a l

I n ( 2 ) he showed u p t h e c o n n e c t i o n

between n f ' s and f i x e d - p o i n t

f r e e permutation groups.

O r e (l) F, urtwangler-Taussky

(1) and T a u s s k y (1) s t a r t e d

a x i o m a t i c c o n s i d e r a t i o n s i n t h e t h i r t i e s f o r w h a t we n o w c a l l near-rings.

A f i r s t name f o r t h e s e s t r u c t u r e s was p r o p o s e d i n 1 9 3 8 b y G I i e l a n d t ( 1 ) : "Stamm"

( = t r i b e ) ("stem"

i s s t i l l used

i n t h e I t a l i a n l i t e r a t u r e ) . W i e l a n d t a l s o announced s t r u c t u r e - t h e o r e t i c r e s u l t s i n t h i s note. The f i r s t o n e s t o u s e t h e n a m e l ' n e a r - r i n g " w e r e

i n 1936 and B l a c k e t t and P . J o r d a n

Zassenhaus

i n 1950.

I n 1 9 3 2 F i t t i n g (1) c h a r a c t e r i z e d t h o s e a u t o m o r p h i s m s o f ( n o n - a b e l i a n ) g r o u p s , w h o s e sum i s a n a u t o m o r p h i s m , thereby i m p l i c i t e l y s t a r t i n a t o consider dgnr's.

too,

13


Finally,

the fifties

b r o u g h t t h e s t a r t of a r a p i d

development o f t h e t h e o r y o f n e a r - r i n g s . Now we a r e g o i n g t o d e f i n e t h e a n a l o g u e o f t h e c o n c e p t o f a module i n r i n g t h e o r y :

2.) 1.17

c e r t a i n o p e r a t o r groups.

N-GROUPS DEFINITIONS L e t

.

(rs+)

be a g r o u p w i t h z e r o o and l e t

NE~.

Nxr-r ( r , u ) i s c a l l e d an N-group ( n 3Y )+nY ("near-module o v e r N" ( b u t c f . t h e d i f f e r e n t meaning e.g.

Let

p:

(1) ) ) i f

i n Karzel-Pieper

V

W

YEr

: ( n t n ' ) y = ny+n'y A (nn')y

n,n'EN

Nr f o r

t h e N-group

t h e c l a s s o f a l l N-groups.

To s i m p l i f y

If t h e m e a n i n g o f p i s c l e a r we w r i t e above.

Let

Ng be Nr

the notation,

further notice.

= n(n'y).

stands f o r N-groups throughout,

without

See a l s o K u z ' m i n ( 1 ) .

1.18 EXAMPLES ( a ) l e t N be a n r . Then an N-group,

u:

NxN-N (n,n')+nn'

makes

(N,+)

into

denoted by

( b ) Each ( l e f t ) module M o v e r a r i n g R i s an R - g r o u p . (c) Let

r

with

b e a g r o u p . Then p:

r

i s an M ( r ) - g r o u p

M(r)xr+r (fsY)

+

1.19 P R O P O S I T I O N T a k e Nr E

oY

f(Y)

Ng.

= 0;

(a)

WyEr;

(b)

b(ycr

b'ncN:

(c)

QnEN,:

no =

(d)

VyEr

(-n)y

= -ny;

0;

bnENc: ny = n o . ..

( a ) a n d (b): a s f o r ( r i n g - ) m o d u l e s . ( c ) : n o = nOo = 00 = o . ( d ) : ny = nOy = n o .

Proof.

M(T) I-

01

14

1.20


DEFINITION NT

W

E

Ng i s

called unitary i f NE

n,

and

ycr: iy = y.

n,

mc,

no, a n d a l l Ng a r e v a r i e t i e s i n t h e s e n s e o f Since u n i v e r s a l a l g e b r a i t makes s e n s e t o s p e a k a b o u t a l o t o f t h i n g s ( s e e a l s o Prehn ( 1 ) - ( 3 ) ) : 3 . ) SUBSTRUCTURES

1.21 DEFINITION ( a ) A subgroup M o f a nr. N w i t h subnear-rinq of N ( n o t a t i o n :

M.Mr M

( b ) A s u b g r 0 u p . A o f Nr w i t h N A c A +) N - s u b g r o u p of r (ASNr). 1 . 2 2 EXAMPLES ( a ) No and N c a r e s u b n e a r - r i n g s is a from 1.13 t h a t ( N , + ) i t s s u b g r o u p s ( N o , + ) and f o r the c o n v e r s e problem o f o u t o f a z e r o - s y m m e t r i c and

is called a

MSN).

i s s a i d t o be a n

o f N . Hence i t f a l l o w s split extension o f (Nc,+). See P i l z ( 9 ) , ( 1 0 ) constructing oear-rings a constant one.

( b ) I f N r i s a ( r i n g - ) module t h e n t h e N - s u b g r o u p s a r e j u s t t h e submodules o f r . L a t e r on we w i l l s e e t h a t t h e s u b n e a r - r i n g s o f t h e M ( r ) ' s a r e i n a c e r t a i n s e n s e a l r e a d y a l l n e a r - r i n g s . We know a l r e a d y o n e p r o c e d u r e t o g e t s u b n e a r - r i n g s o f M(r): t h e M S ( r ) ' s o f 1.4. Two more m e t h o d s a r e : 1.23 EXAMPLES ( a ) Take a s u b g r o u p A o f r . a subnear-rinq of M(r).

MA(r ): = { f c M ( r ) I f ( A )

( b ) T a k e a normal s u b g r o u p A o f r . Mr,*(T): = ( f c M ( T ) ( t / YET: f ( y + A ) C f ( y ) + A l s u b n e a r - r i n g o f M(r) ( c f . B e t s c h ( 3 ) ) . + ) T h e term "N-subRroup

of N" refers t o

".

E

i s a

A)

i s

15


1.24 REMARK W i e l a n d t ( 3 )

proposed a c o n s t r u c t i o n method f o r

o f M ( T ) w h i c h g i v e s t h e 3 k i n d s o f scrbnear-

subnear-rings

r i n g s m e n t i o n e d above as s p e c i a l cases. The m e t h o d i s a s f o l l o w s :

ra

Take a n y c a r d i n a l number a , f o r m t h e d i r e c t p r o d u c t

ra .

and a subgroup A o f t o be Let

Ma,A(r):

( a ) MA(r)

Each

fEM(r)

can be c o n s i d e r e d

i f i t i s d e f i n e d component-wise.

cM(ra)

= IfEM(r)[f(A)

E

A1

5

M(r).

Then

= Ml,A(r)

4 . ) HOMOMORPHISMS A N D I D E A L - L I K E SUBSETS 1 . 2 5 D E F I N I T I O N L e t FL,N'

N+N'

( a ) h:

m,nEN: (b) h:

NryNr'EN9.

i s c a l l e d a ( n e a r - r i n g ) homomorphism h(m+n)

Nr+Nr'

'j y , 6 E r

b e ~ r a)n d

h(m)

+

h ( n ) A h(mn) = h ( m ) h ( n ) .

i s c a l l e d an N-homomorphism

b

ncN:

h(y+6) = h(y)

+

if

h(6) A h(ny) = nh(y).

T h e r e seems t o b e n o n e e d f o r e x p l i c i t

definitions of

nr.-monomorphisms

HomN(r,r'),

K e r h,

I m h,

(Nw N ' ) ,

if

Hom(N,N'),

r 2 Nl"y

I f t h e r e e x i s t s a r n o n o m o r p h i s m Nw N '

and so on.

we s a y t h a t N i s e n b e d d a b l e i n N '

and w r i t e N C

N'. A s i m i l a r

c o n v e n t i o n a p p l i e s t o N-groups. 1 . 2 6 EXAMPLE

For a l l

y ~ ~ hy: r : N+r

E

HomN(N,r)

n+ny 1.27 DEFINITION L e t N

E

a~ n d

NrENq.

( a ) A normal subgroup I o f ( I 9 N) i f a ) IN E I

B)

tl

(N,t)

i s c a l l e d i d e a l of

N

n,n'~N WiEI: n(n'+i)-nn'EI. (N,+) with a ) are called r i q h t Normal subgroups R o f

01

16

T H E E L E M E N T A R Y T H E O R Y OF N E A R - R I N G S

i d e a l s o f N ( R 4, N ) , w h i l e normal s u b g r o u p s L o f ( N , t ) w i t h B ) a r e s a i d t o be l e f t i d e a l s ( L N). ( b ) A n o r m a l s u b g r o u p A o f r i s c a l l e d i d e a l o f Nr ( A dN r ) i f W Y E r W ~ E A W n E N : n(y+6)-nyEA O t h e r n a m e s : N - k e r n e l o r s u b m o d u l e ( c f . 1.33!). The t e r m " i d e a l " i s m o t i v a t e d by ( K u r o s h ) a n d i s v e r y handy i n f o r m u l a t i n g s i m u l t a n e o u s s t a t e m e n t s a b o u t N - g r o u p s arid n e a r - r i n g s

.

.

".

1 . 2 8 R E M A R K S The l e f t i d e a l s o f N c o i n c i d e w i t h t h e i d e a l s o f M o r e o v e r , o n e e a s i l y s e e s t h a t a s u b g r o u p I o f N (A o f r ) i s an i d e a l i f f n1 E n i (mod I ) n 2 E n; (mod I ) => n l t n 2 E n i + n i (mod I ) A n 1 n 2 z n i n h ( m o d I ) y 1 E y i (mod A ) ( y 2 z y i (mod A) A

WnEN:

nyl

Z

=>

y 1 t y 2 z y i t y i (mod A ) A

nyi(mod A ) ,

respectively).

S o : (mod I o r mod A) i s a " c o n g r u e n c e r e l a t i o n " ( c f . ( G r a t z e r ) ) i f I ( A ) i s a n i d e a l . I f I 9 N and I N, we w r i t e I 4 N , e t c . I n 1 . 2 7 , ( a ) B ) a n d ( b ) c a n a l s o be w r i t t e n a s vn,n'EN WisI : n ( i t n ' ) - n n ' ~ I and iycr W ~ E A VncN : n(&+y)-nycA

.

F a c t o r n r ' s N/I ( I sl N) a n d f a c t o r 14-aroups r / A ( A gN r ) a r e d e f i n e d a s u s u a l ( c f . a n y book on u n i v e r s a l a l g e b r a ) . I f L dQ N , t h e n N / L i s m e a n t i n t h e s e n s e o f N-groups. C l e a r l y I01 and N a r e i d e a l s o f N a s w e l l a s { o l and r a r e ones o f N T . These i d e a l s a r e c a l l e d t h e t r i v i a l i d e a l s .

17


1 . 2 9 THEOREM ("Homomorphism t h e o r e m " ) . (a) If

I 9N

t h e n t h e c a n o n i c a l map So N / I

nc-epimorphism. (b) Conversely, Ker h 9 N

if

h:

NU"

N/Ker

and

TI:

N+N/I

i s a

i s a h o m o m o r p h i c i m a g e o f N. i s an epimorphism t h e n % =

N'

.

The c o r r e s p o n d i n g s t a t e m e n t s h o l d f o r N-groups. The p r o o f i s analogous t o t h e one f o r g r o u p s , r i n q s o r universal algebras,

and hence o m i t t e d .

So i d e a l s a r e j u s t t h e k e r n e l s o f ( N - )

homomorphisms.

As u s u a l f o r " s o p h i s t i c a t e d " a l g e b r a i c s t r u c t u r e s we g e t w i t h t h e usual p r o o f :

1 . 3 0 THEOREM ( s o - c a l l e d " Z n d i s o m o r p h i s m t h e o r e m " ) N+N' b e a n e p i m o r p h i s m . T h e n h i n d u c e s a 1-1L e t h: c o r r e s p o n d e 11 c e b e t w e e n t h e subnear-rings

(ideal s) o f

N containing Ker h the subnear-rings

and (ideals)

N) * h(A):

o f N' by

A(E

Moreover,

f o r a l l i d e a l s I o f N containing Ker h

we g e t

N/I 2 h(N)/h(I).

If

n : N+N/I

i s t h e canonical epimorphism,

we t h e r e f o r e

get f o r a l l ideals J o f N containing I

Again t h e analogous statements h o l d f o r N-groups.

Observe

i n t h i s c a s e t h a t f o r t h e l a s t f o r m u l a we h a v e t o a s s u m e t h a t J i s a l s o a n N - g r o u p t o make (cf.

1.33,

1.34).

J/Im e a n i n c r f u l

01

18


1 . 3 1 DEFINITION A s u b n e a r - r i n g M o f N i s c a l l e d i n v a r i a n t i f MNZM and NMEM .

I n v a r i a n t s u b n e a r - r i n q s and i d e a l s c o i n c i d e i n r i n g s , b u t n o t i n n e a r - r i ngs: 1 . 3 2 PROPOSITION

( a ) N o 4, N , b u t n o t g e n e r a l l y

N o It N .

( b ) N, i s a n i n v a r i a n t s u b n e a r - r i n g o f N , b u t i n g e n e r a l n e i t h e r a r i g h t nor a l e f t i d e a l . and Proof. ( a ) No i s a l e f t ideal: f o r a l l n , n ' E N we h a v e ( n + n o - n ) O = nOtnoO-n0 = 0 , so noEN 0 = n(n'O+noO)-nn'0 = n+no-ncNo , a n d [ ( n ( n ' + n o ) - n n ' ) ] O 0 , h e n c e n ( n ' + n J - n n ' ~ N 0' N o i s n o t n e c e s s a r i l y an i d e a l : N:= M(1R) , i d I R c N0 = l = l & M o ( IR) = Mo(IR) , 1: IR+ IR E M ( I R ) , b u t i d ox + l ( b ) N, i s a n i n v a r i a n t s u b n e a r - r i n g :

-

.

tj

ncEN

= ncn

C

,

( n n c ) O = n n C and ( n c n ) O = n c O = w h i c h i m p l i e s t h a t nnccNc a n d n c n E N c . :

nEN

N, i s n o t a l e f t o r r i g h t i d e a l i n g e n e r a l , s i n c e N, i s n o t a l w a y s a normal s u b g r o u p o f ( N , t ) : Take a n o n - a b e l i a n group r a n d y , 6 ~ r w i t h y+6 6+y r + r E M C ( r ) . Now ( i d t f Y - i d ) ( o ) = y,

+

-

.

( i d t f y - i d ) ( 6 ) = 6 + y - 6 $; y i m p l y i n g t h a t id+fy-id&Mc(r) So Mc(r) i s n o r m a l i f f r i s a b e l i a n .

but

.

1.33 R E M A R K I n g e n e r a l t h e r e i s no d i r e c t c o n n e c t i o n b e t w e e n N - s u b g r o u p s a n d l e f t i d e a l s , a s we h a v e seen a b o v e . T h i s i s t h e r e a s o n f o r a v o i d i n g t h e terms " n e a r - m o d u l e s " a n d " s u b m o d u l e s " : s u b m o d u l e s would n o t be n e a r - m o d u l e s i l l g e n e r a l , f o r i d e a l s o f N-groups a r e n o t n e c e s s a r i l y N-subgroups. So i n g e n e r a l N-groups a r e not"R-groups" ("groups w i t h m u l t i p l e o p e r a t o r s " ) i n the sense o f (Kurosh) o r ( H i g g i n s ) . T h i s does n o t happen f o r z e r o symmetric n e a r - r i n g s ( s e e a l s o Prehn ( 1 ) - ( 3 ) ) :

19


1.34 PROPOSITION

(a) L 9, N

NoL

->

E

L

(b) N = No (A dN r Proof.(a)

N

L 9,

(b) ->:

A sN

~ L E L

by (a) {O} 9, L

r ) for

all

r E Nij+

w n o ~ N o : not = no(OtL)-noOEL.

10) SN N => NO = {o) ->

->

N = No

.

(c) is settled similarly. 1.35 PROPOSITION

(a)

NY SN

yENr:

(b)wAsNI':

r.

N o = NC o F A .

is the smallest under all N-subgroups o f F,r. Throughout t h i s monograph w e will w r i t e

So No

NO Of

course, YET:

N =

R = NCy

.

=

NCo = : R

No implies R = { o l . Also, N, .

. By l.l9(d),

1.36 DEFINITION

(a) N(Nr) (b)

i s simple:

Nr is called N-simple: e x c e p t R and r

N(NT)

has no non-trivial ideals.

Nr has no N-subgroups

(cf. 1.35).

is simple then all (N-) homomorphic 1.37 PROPOSITION If N(Nr) images a r e (N-) isomorphic either to (0) o r t o N({oI o r r ) . Proof: by 1.29.

8 1 THE ELEMENTARY THEORY OF NEAR-RINGS

20

1 . 3 8 EXAMPLES ( a ) I n 5 7 we w i l l s e e t h a t s i m p l e nr,'s

(7.30,

M(r) ( I r l

> 2) and

Mo(r) are

7.33).

( b ) S e e B l a c k e t t ( 4 ) f o r some m o r e e x a m p l e s o f s i m p l e n r ! s o f real functions. (c) If

No

N =

then N - s i m p l i c i t y implies (by 1.34(c))

s i m p l i c i t y f o r each

NI"Nq.

S i n c e {O) i s a l w a y s m i n i m a l i n t h e s e t o f a l l i d e a l s o f N ,

we d e f i n e m o r e i n t e r e s t i n g o n e s t o b e m i n i m a l : 1.39 DEFINITION A m i n i m a l i d e a l o f N i s a n i d e a l w h i c h i s minimal i n t h e s e t o f a l l non-zero ideals. one d e f i n e s minimal r i c l h t i d e a l s , (minimal under a l l N-subgroups Dually,

l e f t i d e a l s , N-subgroups

.

$. D), e t c .

one g e t s t h e concepts o f maximal i d e a l s e t c .

%r

1 . 4 0 PROPOSITION I d N i s m a x i m a l i n N ( A i f f N / I (I'/A) i s simple. Proof:

Similarly,

.

i s maximal i n

1.30.

Near-rings i n which every (one- o r two-sided) maximal a r e s t u d i e d i n F e r r e r o - C o t t i

ideal # (01 i s

- Rinaldi (1),(2).

5 . ) ANNIHILATORS

We w i l l n e e d t h e " n o e t h e r i a n q u o t i e n t s " q u i t e f r e q u e n t l y : 1 . 4 1 DEFINITION L e t (A1

: A2):

= {nENlnA2

Abbreviations: (A:&),

(o:A)

be subsets of

A1,~*

({6)

e

All

.

Nr

: A2) = : (6 : A2)%

E

Ng .

similarly for

(&:A).

i s c a l l e d t h e a n n i h i l a t o r o f A.

I f n e c e s s a r y , we i n d i c a t e t h e n r . N i n v o l v e d b y w r i t i n g ( A 1 : A2)N.

r)

21


1 . 4 2 PROPOSITION N o t a t i o n a s a b o v e . I f Al i s a subgroup (normal subgroup, N-subgroup, ( A , : A 2 ) i n ". o f N r , t h e same a p p l i e s t o

ideal)

The p r o o f i s e a s y and t h e r e f o r e o m i t t e d . 1.43 COROLLARY (a)

(b)

b b

:

YEr

A

( 0 : ~ A, )

sNr : ( o : A ) A N

1 . 4 4 PROPOSITION Let

(Ai:A) =

(.(I

n

hi

(

E

A,Ai

1EI

iE1

n

N

: A)

Ai

: A)

(isI) and

be s u b s e t s of Nr. Then (Ai:A) 5 ( A i :A). iE1 is1

u

u

and s i m i l a r l y for the u n i o n .

iEI

1 . 4 5 P R O P O S I I X Let A b e a s u b s e t o f

(a)

( 0 : ~ = )

n

Nr

E

N9.

(o:a)

~ E A

(b)

Nr

'2,

N r b -2

( 0 : r ) = (0:r')

Proof: straightforward. Consider t h e h y ' s o f 1.26. 1 . 4 6 PROPOSITION Ker h y = ( o : y )

,

so

Ny

c

N/(o:y).

P r o o f : homomorphism t h e o r e m .

Nr i

1 . 4 7 DEFINITION 1 . 4 8 PROPOSITION

Nr

s called faithful i f

f a i t h f u l =>.

N

4

phism

h: N

n

with

+

M(r)

+

fn

= {O?.

M(r).

P r o o f : C o n s i d e r f o r e a c h neN t h e map Then

(o:r)

f,,:r+r

.

fncM(r). r*ny t u r n s o u t t o b e a n e a r - r i n g homomot-

Ker h = { n E N J f n = 6 ) = { n E N ( n E ( o : r ) ) = {Ol.

So h i s a n e m b e d d i n g map.

31

22


Nr

1 . 4 9 PROPOSITION Let

(a) If

r

be f a i t h f u l .

i s a b e l i a n then s o i s N .

v

then

.

(b) If

tj

Proof:

( a ) by 1 . 4 8 a n d ( b ) b y a s t r a i g h t f o r w a r d c a l c u l a t i o n .

nEN

: n ( y t 6 ) = ny+n6

y,6Er

nENd

More g e n e r a l l y o n e c a n p r o v e t h a t , i f N r i s f a i t h f u l , e v e r y " i d e n t i t y which h o l d s i n r " ( c f . ( G r a t z e r ) ) a l s o " h o l d s i n N " .

1 . 5 0 PROPOSITION Let (by 1.48). ( a ) R = Io)

(b) R =

r

f a i t h f u l . We a s s u m e t h a t

= (0)

N, N,

Nr be

% -N

= M c ( r )

Then So

t a k e some

3nccNc : n c ( o ) =

N,

nc(y) =

.

meMc(r).

m ( o ) =:yo.

yo.

: m(y) = m ( o ) = y o = n c ( o ) = n c ( y ) , a n d N, = Mc(r).

vyEr

m = nc

If

r,

=

M(r)

N c'

.

R =

F

NEn,,;

P r o o f : ( a ) I f R = { o l then V YEr V nceNc: = n c ( o ) = o = a ( y ) , s o n c = 6. I f N, = {Ol t h e n N = No€%, I f NEBO t h e n R = ( 0 1 by 1 . 1 9 ( c ) (b) If

N

Mc(r) t h e n t h e map

h:

i s an

Nc+T

nc+nc

hence

(0)

N-isomorphism.

zN

r N, Finally,if an a r b i t r a r y yEr

.

by some N - i s o m o r p h i s m h , t a k e h ( y ) =: ncEN,. Then h ( y ) =

= nC = ncnc = nch(y) = h ( n c ( y ) ) = h ( n c ( o ) ) .

So

y = nc(o)

E

n and

R =

r.

See a l s o F e r r e r o - C o t t i ( 7 ) a n d S c o t t ( 1 5 ) .

23


6.) GENERATED OBJECTS

1 . 5 1 PROPOSITION ( a ) The s e t s o f a l l i d e a l s ( r i g h t i d e a l s ,

N-subgroups,

l e f t ideals,

invariant subnear-rings)

form inductive

M o o r e - s y s t e m s o n N.

( b ) The s e t s o f a l l i d e a l s (N-subgroups) f o r m i n d u c t i v e Moore-systems on r .

of

an N-group

(...)

Hence i t makes s e n s e t o speak a b o u t t h e " i d e a l

Nr

generated

by a s u b s e t " . 1 . 5 2 PROPOSITION ( S c o t t ( 6 ) ) L e t

RcN

with

RNER

.

Then t h e

l e f t i d e a l LR generated b y R i s an i d e a l . P r o o f : RN e R c L R b y 1.42,

that

,

so

R

LR c (LR:N)

E

.

(LR:N).

Since

Therefore

( L R : N ) qQ N

LRN c L R

showing

L R A N.

See a l s o 2 . 1 6

and 9 . 1 7 4 .

1 . 5 3 THEOREM

(Beidleman (1))

( a ) If N i s f g . ( 0 . 3 ) a s a n i d e a l ( c f . Van d e r W a l t ( 2 ) , ( 3 ) ) ( e . 9 . i f N ~ 3 , t)h e n e a c h i d e a l ( r i g h t i d e a l , M - s u b g r o u p ) different

f r o m Pi i s c o n t a i n e d i n a m a x i m a l o n e .

(b) I f NT i s f.g.

a s i d e a l w i t h N €71, t h e n e v e r y p r o p e r i d e a l N r i s c o n t a i n e d i n a maximal o n e .

ideal ( N - s u b g r o u p ) o f

Proof ( f o r i d e a l s I o f N ) . L e t N b e g e n e r a t e d b y ( s a y ) z:= IIIIaN). T a k e some c h a i n 1 1 C 1 2 E ... xl, ...,x k . o f e l e m e n t s o f $. I : = IJ In9 N by 1 . 5 1 ( a ) . nc IN If

I=N

then a l l o f SEIN w i t h x l ,

some

xl, ...,x k c I . ...,x k € 15 ' B u t

Hence t h e r e i s then

I s = N,

a contradiction. So (r,Z)f u l f i l l s t h e h y p o t h e s i s o f Z o r n ' s Lemma (unless N = (01, a t r i v i a l c a s e ) a n d c o n s e q u e n t l y c o n t a i n s a maximal element.

If

N

E??~,

one proceeds as i n r i n g t h e o r y .

24

$ 1 T H E E L E M E N T A R Y THEORY OF N E A R - R I N G S

b ) CONSTRUCTIONS

1.) PRODUCTS, DIRECT SUMS A N D S U B D I R E C T P R O D U C T S

-

F o r 1.54

1.62 c f . each book on g r o u p s ,

a l g e b r a . We c i t e e . g . 1.54

DEFINITION l e t

X

iE1 and

(Ni)icI

be a f a m i l y o f n e a r - r i n g s .

w i t h t h e component-wise d e f i n e d o p e r a t i o n s

Ni "

.

'I

(iEI).

e l e m e n t s w i t h a l l components

8

Ni

equal t o zero,

'I+"

of t h e

iE1

1.55 D E F I N I T I O N The s u b n e a r - r i n g o f

INo-

ll Ni

i s called the d i r e c t product

n e a r - r i n g s Ni

E

rings o r universal

from (Gratzer).

-

ll Ni

ieI

c o n s i s t i n g of

those

e x c e p t a f i n i t e number

i s c a l l e d t h e ( e x t e r n a l ) d i r e c t sum

o f t h e Nils.

ic1

ll N.i

More g e n e r a l l y ,

every subnear-ring N o f

p r o j e c t i o n maps

xi ( ~ E a r e Is) u r j e c t i v e ( i n o t h e r words, : ni i s t h e i - t h c o m p o n e n t o f some e l e m e n t

~ E I \ niENi

where a l l

ieI

o f N) i s c a l l e d a s u b d i r e c t p r o d u c t o f t h e Ni's. The d e f i n i t i o n s o f p r o d u c t s , d i r e c t sums a n d s u b d i r e c t p r o d u c t s o f N - g r o u p s s h o u l d be c l e a r now ( f o r d i r e c t s u m s YOU need N = N o ) . A g a i n we r e f e r t o P r e h n ' s p a p e r s . 1.56

N O T A T I O N I f t h e Ni ( i E I ) a r e a s a b o v e , l e t fii A, : = { ( . ,0,ni ,o, ) I nic:Ni I .

..

...

b e g i v e n by

25

1 b Constructions

1 . 5 7 P R O P O S I TION (a)

b

- N. 3 -

i c I : N~

A N i G

8

ic1

J

4 Ji Ni

Ni

N.1

$ N . A N i 9

jcI (b)

A

1

8

j EJ

1 . 5 8 REMARKS ( c f .

N~ 4

A

n N j ; jcI

8

N~ A

i€ 1

(ieI)

n N~

4

jcJ

(Gratzer)).

near-rings Ni

n j cn I N J.

8 N A Ti jc1 j

;

ic1

( c ) J ~ I->

51

n

N~

.

ic1

I f N i s a subdirect product o f

then the

Nils

a r e homomorphic

i m a g e s o f N ( u n d e r t h e p r o j e c t i o n maps n i ) . If K e r vi =:Ki o f ideals o f N with zero i n t e r we g e t a f a m i l y (Ki)icI section. Conversely,

[I

with

if a f a m i l y of

o f som? n r . N i s given then N i s isomorphic t o a

= {Ol

Ki

icI

ideals

subdirect product o f the near-rings Of

course,

1.56

(Ki)icI

= N/Ki.

Ni:

- 1.58 can be t r a n s f e r r e d t o N-groups i n t h e

o b v i o u s way. 1 . 5 9 D E F I N I T I O N A s u b d i r e c t p r o d u c t N o f n e a r - r i n g s Ni ( i c 1 ) i s called t r i v i a l i f 3 i c I : ni i s an isomorphism. NEW i s c a l l e d s u b d i r e c t l y i r r e d u c i b l e i f N i s n o t isomorphic t o a n o n - t r i v i a l subdirect product o f nearrings. T h e same i s d e f i n e d f o r N - g r o u p s . 1 . 6 0 T H E O R E M ( ( G r a t z e r ) , F a i n ( 1 ) ) . The f o l l o w i n g c o n d i t i o n s

f o r a nr. (a)

N

{O)

N

( b ) If ( I a ) a c * I a = (01

n

acA

(c

1

are equivalent:

i s subdirectly irreducible;

n

{01+14N

I

+

i s a f a m i l y o f i d e a l s of N w i t h then

3

aEA

: I a = CO);

{UI;

( d ) N contains a unique minimal ideal, o t h e r non-zero i d e a l s .

contained i n a l l

81

26


Replacinq " N " N-groups.

by 'INTI' y i e l d s a n a n a l o q o u s t h e o r e m f o r

1 . 6 1 C O R O L L A R Y Each s i m p l e n r . ( N - g r o u p ) i r r e d u c i bl e .

i s subdirectly

1 . 6 2 THEOI?EM ( ( G r a t z e r ) , p.124). ( a ) Each n e a r - r i n g i s i s o m o r p h i c t o a s u b d i r e c t p r o d u c t of subdirectly irreducible near-rinqs.

( b ) Each N-group i s N - i s o m o r p h i c t o a s u b d i r e c t p r o d u c t o f s u b d i r e c t l y i r r e d u c i b l e N-groups. The i n t e r s e c t i o n o f a l l n o n - t r i v i a l in ilartney ( 3 ) .

ideals i s a l s o considered

2 , ) N E A R - R I N G S OF Q U O T I E N T S 1 . 6 3 DEFINITION L e t N be a n r . a n d S a s u b s e m i g r o u p o f ( N , . ) . A n e a r - r i n g N S i s c a l l e d a n e a r - r i n g of riclht ( l e f t ) q u o t i e n t s of N w.r.t. S i f ( a ) NS

E n ,

( b ) N a N S ( b y h, s a y )

( c ) v SES: h ( s ) (d)

qENS

3

is invertible in

SES

3

nEN

(NS,v)

: q = h(n)h(s)-'

(q = h ( s ) - l h ( n ) ) .

Of c o u r s e t h e r e a r i s e t h e q u e s t i o n s a b o u t e x i s t e n c e a n d u n i q u e ness o f s u c h n e a r - r i n g s o f q u o t i e n t s . We w i l l s e t t l e t h e s e questions a f t e r the following 1 . 6 4 DEFINITION N i s s a i d t o f u l f i l l t h e r i q h t ( l e f t ) @ r e c o n d i t i o n (Ore ( 1 ) ) w . r . t . a g i v e n s u b s e m i g r o u p S o f (N,.) if

V

(s,n)

E

SxN

3

(sl,nl)

E

SxN

:

nsl = s n l

(sln = nls)

1b Constructions

1 . 6 5 THEnREM ( G r a v e s - M a l o n e

N

L e t S be a subsemiaroup of

(1)).

I

(N,.).

27

has a n r . o f r i g h t q u o t i e n t s w . r . t .

(a)

s

(b)

v

S

+0

SES : s i s c a n c e l l a b l e (on b o t h s i d e s )

( c ) N s a t i s f i e s t h e l e f t Ore c o n d i t i o n w . r . t . A s s u m e t h a t N h a s a n r . NS o f

P r o o f . ->:

w.r.t. (a):

S,

le f t

S.

quotients

and l e t h b e as i n 1.63.

By 1 . 6 3 ( d )

(b):b

SES

m,nEN

h ( m ) = h ( n ) ->

->

Take

= h(n)h(s)->

m = n.

q: = b ( s )

scS.

nEN,

h(m)h(s)

.

m = n

sm = s n ->

Similarly, (c):

: ms = n s =>

-1 h(n)

,

N,

E

so b y

1.63(d) 3nlcN

: h(s)-'h(n) h(nsl) = h(snl)

]sics

Therefore

,

(n,s)%(n',s'):

3nlEN

nsl = n'nl). H.J.Weinert

n 'n ~

SCl

%

on NxS

slES

by

: ( s s l = s ' n l and

b e t h e equivalence c l a s s o f (11,s) and we m i g h t f o l l o w a s u g g e s t i o n o f n'

I f s,7~NS,

_

p. 262):

Jacobson),

t o clet "common d e n o m i n a t o r s " :

5s

=

3 S S I '

37

w i t h ( n , , ~ , ) E N ~ Sf u l f i l l i n n s ' n ,

_

=

ssl E S

ss 1

!Je t h e n a r e a b l e t o d e f i n e w i t h t h e s e n o t a t i o n s :

.-

n n' + 7.-

s

ns t n ' n 1 1

s

ss 1

w h e r e (n,,~,) t

and

E

and

n n' '"2 s-7:= I,

s2 llxS f u l f i l l s n s 2 = n ' s 2 E S .

a r e shown t o b e w e l l - d e f i n e d

turns o u t t o be a nr. with i d e n t i t y element o f 5 ) .

If

t E S ,

t h e map

i s a mononorphism a n d e v e r y inverse

st'

=

I

n'n, 1_-

n

n n '

NS.

IlxS/% =:

-

Let

3

.

n s l = snl

whence

(N.

1x1

=

IX'I

.

i s s e t t l e d by 1 . 7 1 .

n

( N ~ c)o n t a i n f i n i t e s t r u c t u r e s w i t h more t h a n

one e l e m e n t ( f o r n r l s e . g . t h e f i e l d Z+, N-groups e . g . (Z,,+) w i t h nO = n l = 0 n & N ) . Now a p p l y ( G r a t z e r ) , p . 1 9 7 .

for for all

1.76 R E M A R K Note t h a t t h e theorem above a l s o h o l d s e . g . f o r a f r e e n r . (N-group) i n t h e v a r i e t y of a b e l i a n nr!s ( N groups) ( " f r e e a b e l i a n nr!s (N-groups)").

1 . 7 7 R E M A R K Let be f r e e o v e r X . The usual c h a r a c t e r i z a t i o n ( i n t h e c a s e o f u n i t a r y ( r i n g - ) modules) o f X a s a b a s e ( ' l i n e a r l y i n d e p e n d e n t g e n e r a t i n g s e t " ) does not c a r r y o v e r t o t h e c a s e of N-groups d i r e c t l y : N-groups d o n o t have t o be u n i t a r y , t h e l a c k o f c o m m u t a t i v i t y i n N@ c a u s e s " l i n e a r c o m b i n a t i o n s " ( d e f i n e d as u s u a l ) t o be i n f l u e n c e d by t h e o r d e r o f t h e summands ( a s Maxson ( 1 ) p o i n t e d o u t , one has t o d e f i n e l i n e a r c o m b i n a t i o n s i n t e r m s o f o r d e r e d a n d - most t r o u b l e s o m e of a l l s e t s of elements of N @ u s u a l l y c o n s i s t s o f more then t h e s e t o f a l l l i n e a r combinations, s i n c e i n qeneral n ( n l y l + . . + n k y k ) i s no l i n e a r combination any more. Anyhow, g e n e r a l i z i n g t h e c o n c e p t o f l i n e a r independence g i v e s something l i k e a b a s e : l e t W n ( n & I N o ) be t h e s e t o f a l l n - a r y words o v e r some s e t X i n a v a r i e t y V o f N-groups a n d ( f o r , r E V ) w r t h e induced f u n c t i o n r n + r . D e f i n e i n W n W % ~ W ' : b A E 0 :wA = w;

-

.

.

31

1 b Constructions

1 . 7 8 DEFINITION A s u b s e t B o f

NrcV

i s c a l l e d independent i f

WrcR: r M = 1 . 7 9 REMARK L e t RM be a u n i t a r y r i n g - m o d u l e w i t h = (01 -> r = 0 ( o t h e r w i s e RF1 w o u l d h a v e n o l i n e a r l y independent subset a t a l l ) . Then each s u b s e t o f RM i s l i n e a r l y i n d e p e n d e n t i f f i t i s i n d e p e n d e n t i n t h e s e n s e o f 1.78.

Nr

1.80 DEFINITION B E ( a ) B generates

i s called a base f o r

Nr i f

Nr

B i s independent.

(b)

As u s u a l , t h e f o l l o w i n g q u e s t i o n s a r i s e : ( a ) Which N-groups have a base ? (b) Are d i f f e r e n t bases e q u i p o t e n t ? 1.81 THEOREM B t:

B

+

r

Nr

E

Nr i f f

i s a base f o r

t h e i n c l u s i o n map

can be e x t e n d e d t o an N - i s o m o r p h i s m

O+r,

where 0 i s t h e f r e e N - g r o u p on B.

-P r o o f .

->:

L e t B be a b a s e f o r

S i n c e 0 i s f r e e o n 6,

with hof = 1 h i s an N - i s o m o r p h i s m : (a) Bth(@) A B generates I#JE@

w(f(Bl),

Consider the diagram

t h e r e i s e x a c t l y one

hEHomN(O,r)

(b) L e t

Nr.

We h a v e t o show t h a t

.

r

=>

h(4)

=

r

b e i n K e r h . R e p r e s e n t I$ b y some w o r d

...,f ( B 6 ) )

over

o = h(O) = h(W(f(B1) = wr(h(f(a1) = w r ( 6 1,...,3,).

f(B)

,..., f(B,)))

,..., h ( f ( B , ) ) )

( B ~ + R ~f o r i + j ) . f . ( o w =

= w,(~(B1)s***s~(B,)) 20 6 = 5 .

B i s inCeq2nient.

=

32

$ 1 THE ELEMENTARY THEORY OF NEAR-RINGS

e x t e n d s t o a n N-isomorphism N @ , f ( B ) i s independent a n d g e n e r a t e s N@. So f ( B ) i s a b a s e f o r N @ . Hence B = I(B) = h ( f ( B ) ) i s a b a s e f o r h ( 0 ) = r . Suppose t h a t

(1.15).

N = Nd ==>

See M e l d r u m ( 1 3 ) f o r a n e x a m p l e

o f a f a i t h f u l , s i m p l e N=No-group

r

(with NEW,)

which is

not unitary.

I t i s sometimes d e s i r a b l e t o l o o k f o r an embeddinq o f N i n t o

M(r)

with a "smaller"

e.g.

M(T)

as above.

R e c a l l t h a t i n 1.86 and 1.87

i s e m b e d d e d i n t o t h e much b i g g e r M ( M ( r ) @ ? $ ) !

For doing t h i s ,

we g e n e r a l i z e a c o n c e p t d u e t o M e n g e r :

1 . 9 1 DEFINITION 6

v

r

n,n'cN:

e

N

i s c a l l e d a base ( o f e q u a l i t y ) i f

( b b c B : n b = n ' b ) .->

1 . 9 2 REMARK C l e a r l y 6 f o r m s a b a s e i f f

n = n'.

(0:B) = I O I , s o i t

w o u l d n o t b e n e c e s s a r y t o u s e a s p e c i a l name. B u t we d o it, because i t i s a v e r y s u g g e s t i v e one.

1 . 9 3 E X A M P L E S I n M(r) t h e s e t Mc(r) ( a g r o u p i s o m o r p h i c t o r f o r m s a b a s e . I n Mcont(r) ( 1 . 4 ( a ) ) i t s u f f i c e s t o t a k e a dense s u b s e t o f

r.

)

l c Embeddings

35

This motivates the i n t e r e s t in the case t h a t t h e constants Nc f o r m a b a s e . T h i s csn b e a c h i e v e d by f o r c e : 1 . 9 4 PROPOSITION Let TI be t h e n a t u r a l e p i m o r p h i s m Then n ( N c ) f o r m s a b a s e f o r n ( N )

.

6 = n(n)

N+N/

(O:Nc)

'

E l = n(nl) are ET(N) then ( V 7 i c ~ ~ ( ~ :c n) n c = Ti 1n c ) - > ( b n c ~ N c : n n c - n 1n c€ ( O : N c ) ) = > =-z ( b n c e N C : 0 = (nnc-nl!c)nc = n n c - n l n c = ( n - n ) n ) => 1 c -> ( n - n l ) E ( O : N c ) -> n = n 1 '

Proof. If

and

--

1 . 9 5 EXAMPLES ( a ) In

M(T),

t h e c o n s t a n t s form a b a s e .

t h e c o n s t a n t s I p do n o t f o r m a b a s e . xp-x p o ( z e r o p o l y n o m i a l ) , b u t Va€Zp : ( x P - x ) ( a ) = ap-a = 0. Therefore xp-x~(O:iZp). (0:Z ) c o n s i s t s o f a l l p o l y n o m i a l s w h o s e c o r r e s p o n d i n q P p o l y n o m i a l f u n c t i o n i s t h e z e r o map.

(b) In

Zp[x], In f a c t ,

The f o l l o w i n g s o l v e s t h e p r o b l e m s t a t e d a f t e r 1 . 9 0 . 1 . 9 6 T H E O R E M I f B sNN

,

the following conditions a r e equivalent:

( a ) B i s a base (of equality);

( b ) B i s a f a i t h f u l N-group; ( c ) N4M(B). P r o o f . 1 . 4 8 and 1 . 9 2 . 1 . 9 7 C O R O L L A R Y I f N, i s a b a s e t h e n N c a n b e c o n s i d e r e d a s a n e a r - r i n g o f f u n c t i o n s on N c . I n v i e w o f 1 . 9 5 ( a ) , t h i s I s "the natural representation o f N". 1 . 9 8 DEFINITION L e t r,A b e g r o u p s . fEM(T) i s c a l l e d k e r n e l if free if y c r : ( f ( y ) = ' o -> y = 0 ) . P u t M ( T ) q k M(A) t h e r e i s some h : M ( r ) w M ( A ) such t h a t h s e n d s k e r n e l f r e e elements of M ( r ) i n t o k e r n e l - f r e e ones o f M(A), a n d M ( r ) Zk M(A) i f M(r) qk M ( A ) by an i s o m o r p h i s m h .

v

36 1.99


01

r,A

THEOREM ( H e a t h e r l y - M a l e n e ( 1 ) ) . L e t

r+A

Then Proof.

M o ( r ) 4 M o ( A ) k

rCA

(a) Let

b y h.

r

If

'I

arbitrary,

ycr, y

If

,

fEMo(r) f y : A

define

(01 , the r e s u l t i s

=

o b v i o u s . Assume t h a t but fixed

(01

a n d t a k e some

Y

&+C

If f i s kernel-free,

embeds Mo(r) i n t o M o ( A )

( A = (01

i s again t r i v i a

+ 0,

a

f

h i s moreover i n j e c t i v e : h ( f a ) = 6 ==> h ( f a ) ( 6 ) = 0 ,

a = o

diction) or

r

Mz(r)aMz(A)

,

SO

and A

so

a

i s kernel-free, (a contra-

6 = 0

= 6. a (as groups). But

whence

a r e isomorphic groups,

M:(A)

1.

( N o t a t i o n as i n 1 4 ( a ) ) i s a

M:(A)

fa + fs(f,)(6) g r o u p homomorphism. I f

Hence

and

Y

b y ( s a y ) 9 t h e n t a k e some

k

+

64Im h

t h e same a p p l i e s t o f

( b f If Mo(r) 4 M o t h )

h : M:(r)

6cIm h

1

h(f(Y)

~ E A *

by

f €MO(A)

h(f(h-'(d)))

fixed

.

o

A

+

t h e map f + f y

be g r o u p s .

M(r)c,M(A).

.

M:(r)

and

a n d t h e same a p p l i e s t o

r4A.

r+A

(c) If

t h e n p r o c e e d i n o as i n ( a ) one sees

M(r)GM(A)

that

.

(d) I f M ( r ) q M ( A ) t h e n MC(r)+Mc(b) by r e s t r i c t i o n . % Mc(r) r and Mc(d) % A i m p l i e s t h a t r - h

-

1 . 1 0 0 COROLLARY ( B e i d l e m a n ( 5 ) )

Neumann ( 1 ) ) ( W n E N 3 h E N : n : n ( - 1 ) = n -> n = 0 ) .

n = 0.

h+h)

A

(b ncN

A

n,n'cN : n t n ' = ( - n ) ( - l ) + ( - n u ) ( - 1 ) = = (-n-n')(-1) = -(-n-n') = nl+n.

Proof.

(a)

(.b ). D e f i n e

a: N

.

N C l e a r l y a c A u t ( N , + ) and n(-1) a 2 = i d . a ( n ) = n i m p l i e s n = 0 . S o by a t h e o r e m o f g r o u p t h e o r y ( e . q . ( W . R . S c o t t ) , ~ . 357), N i s abel i an.

n

-+ -+

( c ) a ( a s above) i s again a f i x e d - p o i n t - f r e e autom o r p h i s m o f o r d e r 2 . From g r o u p t h e o r y ( B . H . Netlmann ( l ) , p . 2 0 6 ) we know t h a t N i s a b e l i a n . 1 . 1 1 0 R E M A R K M c Q u a r r i e ( 2 ) showed t h a t 1 . 1 0 9 ( b ) d o e s n o t h o l d i n the i n f i n i t e case. We now c o n s i d e r c a n c e l l a b l e e l e m e n t s .

1 . 1 1 1 PROPOSITION L e t N b e a n r . ( a ) (Maxson ( 1 ) ) neN i s r i g h t c a n c e l l a b l e a right zero divisor; ( b ) (Maxson ( 1 ) )

nENo

l e f t zero divisor;

is l e f t cancellable

n i s not a nc = n C 0 = (no+n C )O = rOER , s o T h e rest is trivial.

nOER, too.

2.19 REMARKS 2 1 8 does not hold f o r left ideals L of N. All E E L h a v e the form S! = n +n o c with noEN 0 and nc€Nc , but in general no$L and nc$L: Consider N = P [ x ] and L:= I C a i xi 1 C a i ~ 2 =Z I0,+2,+4, 1 1 . L i s a left ideal o f N (even a maximal one - see S o ( l ) ) , but E:= x + ~ E L decomposes a s S! = no+n C with no = x&L .

...

2.20 THEOREM Under forming sums and intersections, the ideals of N (Nr with NEno ) form a complete modular lattice.

Proof. follows from (Kurosh), p . 143. 2.21 REMARK T h e s e lattices are not necessarily distributive. But cf. t h e following considerations and 2.18 (and also Scott ( 3 ) ) . 2.22 PROPOSITION (Scott (4)) If A,B qN r and A , B sN r then \I nEN b a cA b B E B : n(a+B) z na+nB(mod A r r B ) .

Proof. n(a+B)-nB-na E A+A = A So n(a+B) 5 na+nb(mod A). Similarly, n(a+B) z nB+na(mod and t h e result f o l l'ows.

B ) : na+nB(mod

O n e c a n suspect that 2.22 will be particularly important for A n B ( 0 1 : s e e 2.29.

B),

82

48

IDEAL THEORY

2.23 PROPOSITION ( W i e l a n d t ( 2 ) ) . I f

NE??,

and

A,B,A

%r

then rl:

( A + A )0 ( B + A ) / ( A

i s commutative a n d

ncN

B)+A

y1,y2€r'

: n(y1+y2) = nyl+ny2

.

.

( B e t s c h ( 5 ) ) . E : = ( A n B)+A ; H: = ( A + A ) ~ ( B + A ) Let nl,r12€H a n d ncN Then 1 a E A 3 B E B : rll a(mod E ) A n 2 : B(mod E ) . Now a+B I B+a(mod A A B ) a n d n ( a + B ) E na+nB(mod AnB) by 2 . 2 2 . S i n c e A O B t E we g e t and q 1 + q 2 E a+B z B+a I n2+ql(mod E ) and the n(r11+n2) I n ( u + B ) : n a + n @ E nrll+nq2(mod E ) , p r o p o s i t i o n i s proved.

Proof.

.

2 . 2 4 C O R O L L A R Y ( B e t s c h ( 6 ) ) . With t h e a s s u m p t i o n s a n d n o t z t i o n s of 2.23, = N/(o:r,) is a ring.

m:

P r o o f . r' c a n be c o n s i d e r e d a s a f a i t h f u l TI-group i n t h e o b v i o u s way. Now t h e r e s u l t f o l l o w s f r o m 1 . 4 9 .

NET

a n d i f no 2 . 2 5 C O R O L L A R Y ( B e t s c h ( 6 ) ) . I f N€VO a n d n o n - z e r o homomorphic i m a g e o f N i s a r i n a then t h e l a t t i c e of l e f t i d e a l s of N i s d i s t r i b u t i v e . Proof. Let L 1 , L 2 , L 3 be l e f t i d e a l s o f N . C o n s i d e r t h e N - g r o u p r : = ( L 1 + L 3 ) a ( L ~ + L ~ ) / ( LL 2~ ) ~+ ~ 3

If r 9 { o j t h e n (o:r) N, f o r NET$. From 2 . 2 4 we k n o w t h a t N / ( o : r ) i s a r i n g (9N ) , a c o n t r a d i c t i o n . S o r = I01 a n d t h e l a t t i c e o f l e f t ideals i s distributive. F i n a l l y , l a t t i c e t h e o r y p r o v i d e s us w i t h two more l a w s f o r t h e i d e a l l a t t i c e o f a n r . N o r ,,,T ( N E % ) . L e t I , J , K b e i d e a l s . Modular l a w : I f K e I t h e n I n ( J + K) = ( I n J ) + K . C a n c e l l a t i o n law: I f I E J t h e n I n K = J n K , I u K = J u K i m p l i e s I = J ,

49

2a Sums

2.)

2.26

D I S T R I B U T I V E SUMS

DEFINITION

I'

( a ) A d i r e c t sum

Ia =:I

o f ideals

Ia o f

N (aeA)

ib) =

.EAiaih .

aEA

i s called distributive:

1i

W

,

aeA a

( b ) A d i r e c t sum

Nr

BEA

i;

I*Aa aEA

: (

EI

1 ia)( I

aEA

BEA

o f ideals

=:A

i s called distributive:

Aa

of

(aoA)

( N o t e t h a t t h e sums i n v o l v e d a r e a c t u a l l y f i n i t e o n e s ; a l l summands s h o u l d come f r o m d i f f e r e n t i d e a l s . )

2.27

@ Na

N =

EXAMPLES I f

o f the ideals

ma

then N i s t h e d i s t r i b u t i v e sun

aEA

(1.56).

The same a p p l i e s t o N - g r o u p s . 2.28

PROPOSITION Let sum i s d i r e c t .

(Ia)aEb Then aEA

Moreover:

N whose i s distri-

be a f a m i l y o f i d e a l s o f

Ia

8

?,

I*

Ia

Ia

aEA

aEA

b u t i ve.

N

The a n a l o g o u s r e s u l t h o l d s f o r N - g r o u p s w i t h

=

No

.

Proof. obvious. 2.29 P R O P O S I T I O N ( H e a t h e r l y ( 2 ) ) . o f ideals o f

Nl' w i t h

1 Aa

Let

=

aeA

v Conversely,

ncNo

b

C6a~A:

be a f a m i l y

(Aa)aEA I*Aa aEA

n ( E a a ) = En&,

.

i f Nl' i s f a i t h f u l a n d i f f o r

~ C & , E A : n(Z6,)

= CnCa

then

nENo

Then

=:A.

nEN

.

P r o o f . The f i r s t a s s e r t i o n f o l l o w s f r o m 2.22

and b y

i n d u c t i o n . See a l s o 2 . 6 ( b ) .

I f f o r n c N a n d a l l Z 6 a ~ A n(CSa) = n(oto) = notno, h e n c e no = 0. So

tf

yEI' : ( n 0 ) y = n ( 0 y ) = no = o = O y nO = 0.

then and c o n s e q u e n t l y

50

9 2 IDEAL THEORY

From 2 . 2 9 we g e t t h e f o l l o w i n g s a t i s f a c t o r y r e s u l t ( r e c a l l that for

I*Aa

?r uN

N

p

8

ha

No

,

a EA

acA

t h e r e i s no chance a t a l l t h a t a l w a y s f o r the

2 . 3 0 THEOREM ( B e t s c h ( 3 ) ) . if N

a r e n o t n e c e s s a r i l y N-groups).

bats

Each d i r e c t sum o f i d e a l s i n N ( a n d ,

a l s o in N r ) i s d i s t r i b u t i v e .

P r o o f . The -

If

statement f o r

l*Ia =:I

aeA

and

Nr i s c l e a r f r o m 2 . 2 9 .

1 la,1 ii € 1

then

BEA

aaA

b) CHAIN CONDITIONS

2 . 3 1 REMARKS By 1.51, t h e i d e a l s f o r m a n i n d u c t i v e M o o r e system.

I t makes s e n s e t o s p e a k a b o u t t h i n g s l i k e

"the i d e a l s f u l f i l l the DCC" e t c . By 0.10,

i s f.g.

.

i f t h e i d e a l s f u l f i l l t h e ACC t h e n each i d e a l

2 . 3 2 CONVENTION I f t h e s e t o f i d e a l s f u l f i l l s t h e D C C we s a y t h a t " N f u l f i l l s t h e DCC f o r i d e a l s " o r more b r i e f l y t h a t

" N has t h e D C C I " . To s i m p l i f y s t a t e m e n t s , t h e p h r a s e " L e t N have t h e D C C I " w i l l be a b b r e v i a t e d b y " D C C I " . S i m i l a r c o n v e n t i o n s a p p l y t o r i g h t i d e a l s (DCCR),

left

i d e a l s (DCCL) a n d N - s u b o r o u p s (DCCN). O f course,

t h e same i s d o n e f o r t h e A C C .

2.33 REMARK C l e a r l y t h e DCCN i m p l i e s t h e D C C I i f D C C R o r DCCL i m p l y t h e D C C I .

i m p l i e s t h e DCCL. The same h o l d s f o r t h e A C C .

If

N = No

N = N

-

0'

i n N,

t h e n t h e DCCN

51

2b Chain conditions

2.34 E X A M P L E S

( a ) (Beidleman ( 1 ) ) . L e t a g r o u p r c o n t a i n o n l y f i n i t e l y many normal subgroups b u t an i n f i n i t e c h a i n r = = A 1 = A 2 =... o f s u b g r o u p s ( s u c h g r o u p s a r e known t o e x i s t ) . N: = { f e M ( r ) l t l i e l N : f ( A i ) E A i l . has t h e D C C I b u t n o t Then i t i s immediate t h a t 'I,, t h e D C C N ( s i n c e a l l A i sN r ) . (b)

Each r i n g s a t i s f y i n g t h e A C C I b u t n o t t h e D C C I (Z, f o r i n s t a n c e ) o r c o n v e r s e l y i s o f c o u r s e a n example of a nr. w i t h t h e same p r o p e r t i e s .

2.35 T H E O R E M

( a ) I f I A N and N has t h e D C C I ( D C C N , D C C L ) t h e n t h e same a p p l i e s t o J / I .

I 4N

a n d I i s a d i r e c t summand t h e n N has t h e D C C I ( D C C N , D C C L ) I a n d N/I have t h e D C C I ( D C C N ,

(b) If

DCCL), (c) If

A

AN r (NEW,)

i s a d i r e c t sumniand t h e n

DCCI (DCCN) i f f A a n d

r/A

r has t h e

have t h i s p r o p e r t y .

Proof. ( f o r i d e a l s o f N and the DCCI)

...

( a ) Let J l d 2 ? be a d e s c e n d i n g c h a i n of i d e a l s o f N / I . I f J i : = T -1 ( J i ) ( i E I N ) t h e n J 1 3 J 2 3 by 1 . 3 0 . S o 3 neIN V k r n : J k = J n . S i n c e V icIN : n ( J i ) = = l r ( T - 1 ( 3 i ) ) = Ji by 2 . 1 7 ( a ) , Jk = Jn f o r a l l k z n .

...

I t r e m a i n s t o show t h a t I h a s a l s o t h e D C C . B u t t h i s f o l l o w s from t h e f a c t t h a t e a c h i d e a l o f I i s an i d e a l o f N. ( b ) ->:

L e t I a n d N/I have t h e D C C and l e t J 1 3 J 2 ? , . . be a c h a i n of i d e a l s o f N. The c h a i n s J l o 1 3 J 2 n 1 2 ... and a(J1+I)?a(J2tI)?. g e t c o n s t a n t a f t e r some nEIN. Therefore b k 2 n : JknI = J n n I A n ( J k + I ) = = n(Jn+I)

( b ) : t r i v i a l .

( b ) => ( c ) : I f

N =

1 Ia ,

define

aEA

4 :=

J c o n t a i n s a maximal e l e m e n t ( w . r . t . 5 ) B . acA: ( I a n N ' = I a v I a n N ' = {O}) I n n N 1 = {Ol

-

of

B.

( c ) ->

So

'd

i s aEA

.

a c o n t r a d i c t i o n t o the maximality : 1,cN'

a n d hence

N = N1 =

I*IB

€3 E B

*

( d ) : by d e f i n i t i o n .

( c ) =.> ( e ) : I f I 9 PI, c o n s i d e r a n i d e a l J maximal ( Z o r n ! ) w i t h t h e p r o p e r t y t h a t J n I = {Ol. N ' : = I i J . I f N =f N ' , 3 J 0 d : J o s i m p l e A J o ~ N i A J o ~ I O Then l.

JonN' = ( 0 1 ,

so

J+Jo=J.

( J t J o ) n I = (01,

Also,

since x = j t j o c ( J + J o ) n I implies t h a t j o = = x - j c ( I t J ) n J o = N 1 n J o = {Ol. T h i s c o n t r a d i c t s t h e m a x i m a l i t y of J . T h e r e f o r e IGJ = N a n d I i s a d i r e c t summand.

( e ) -> ( a ) : I f I 9 N, d e n o t e by T t h e sum o f a l l s i m p l e i d e a l s o f I. Assume t h a t I I. T 9 I A N A I i s d i r e c t summand => T 9 N Hence T i s i t s e l f a d i r e c t summand a n d t h e r e i s some

-

.

J A N with

TGJ = N.

3 2 IDEAL THEORY

56

C o n s e q u e n t l y each s i m p l e i d e a l of T i s a simple i d e a l o f N . T G ( J n 1 ) = I , s i n c e e a c h iE1 h a s t h e form i = T + j w i t h TET a n d j E J ; because of TSI we know t h a t j c 1 . We n o w show t h a t J n I c o n t a i n s a s i m p l e n o n - z e r o i d e a l of N and a r r i v e a t a c o n t r a d i c t i o n . By a s s u m p t i o n , J n I $. {O). I f J n I i s f.g. then t h e r e e x i s t s a maximal i d e a ? I* i n J n I , a n d e a c h d i r e c t complement ( e x i s t e n c e a s b e f o r e ) o f I* i n J n I i s a s i m p l e n o n - z e r o i d e a l of JnI and o f N. I f J n I i s n o t f . g . , t a k e any f g . i d e a l F I01 o f J n I . T h e n F 9 J n I . F c o n t a i n s a maximal i d e a l M I f ) : S i n c e ( c ) => ( a ) , e v e r y I N i s t h e sum (and by ( b ) => ( c ) t h e d i r e c t sum) o f s i m p l e i d e a l s , i m p l y i n g t h a t I i s completely r e d u c i b l e . I f I d N, t a k e some J g N ( a g a i n , J i s c o m p l e t e l y 2r J by 2 . 8 r e d u c i b l e ) w i t h 1;J = N . B u t t h e n N/I and N/I i s c o m p l e t e l y r e d u c i b l e .

-

( f ) => ( d ) :

trivial (take

( a ) ->

(9): trivial.

( 9 ) ->

( e ) : a s i n ( c ) =>

I = N).

(e).

2 . 4 9 C O R O L L A R Y The d i r e c t s u m o f c o m p l e t e l y r e d u c i b l e n e a r - r i n g s i s again completely r e d u c i b l e . (N-groups w i t h NE??,)

Near-r.ings ( N - g r o u p s ) which decompose i n t o f i n i t e l y many simple i d e a l s a r e e s p e c i a l l y important. The f o l l o w i n g theorem w i l l b e used f r e q u e n t l y t h r o u g h o u t t h i s book. M u c h more o n t h i s s u b j e c t can be found i n B l a c k e t t ( l ) , C h a o ( I ) , Hartney ( Z ) , Oswald ( 3 ) , ( 4 ) , ( 5 ) , ( l o ) , N a t a r a j a n ( 1 ) , Ramakotaiah ( 3 ) and S c o t t ( 7 ) .

57

2c Decomposition theorems

.

2 . 5 0 THEOREM (Beidleman ( l ) , Betsch ( 3 ) ) . Let N be a n r . E q u i Val e n t a r e : ( a ) N i s t h e sum o f f i n i t e l y many s i m p l e i d e a l s . ( b ) N i s t h e d i r e c t sum of f i n i t e l y many s i m p l e i d e a l s . ( c ) N i s c o m p l e t e l y r e d u c i b l e and has t h e D C C I a n d t h e ACCI. ( d ) N i s c o m p l e t e l y r e d u c i b l e and has t h e ACCI. ( e ) N i s c o m p l e t e l y r e d u c i b l e a n d has t h e D C C I . ( f ) N i s completely reducible a n d every ideal o f N i s f ( 9 ) There e x i s t maximal i d e a l s 11, I n of N with zero i n t e r s e c t i o n , b u t a l l J r : = Ik (0). n k+r ( i n t h i s c a s e , N = T'Jr and Jl J n are simple). r =1

...,

n

+

,...,

( h ) There e x i s t maximal i d e a l s

n

n

Il,...,In

with

I,. = {OI.

r =1

The u s u a l changes y i e l d a n a l o g o u s r e s u l t s f o r N-groups w i t h Ncno (remark a l s o t h e a d d i t i o n a l r e s u l t s i n Oswald ( 2 ) ) . P r o o f . ( a )

( b ) : as in 2.48.

n

1 I k ( a l l I k simple) then k=l N i s c l e a r l y c o m p l e t e l y r e d u c i b l e . Moreover, N = I ~ / /I, a I ~ / . . . / I , - ~ a a I~ (01 i s a p r i n c i p a l s e r i e s , so N f u l f i l l s both c h a i n c o n d i t i o n s by 2 . 4 1 . ( b ) =>

(c): If

N =

...

( c ) ->

( d )

( d ) a n d ( c ) ->

...

(e) are trivial.

( f ) : by 0.10.

A C C ( D C C ) f o r c e s A t o be f i n i t e .

$ 2 IDEAL THEORY

58

( b ) -> (9): I f d e f i n e Ik: =

F

...;Jn

N = J1;

r kJ r

I k a r e maximal i d e a l s . I f

3

and i f

(jiEJi)

(Ji

. Because x

E

...,n l :

kE{l,

simple ideals).

of

N/Ik

fi

k=l

p

a1 1

...+j n

x = jl+

Ik,

jk

- Jk, 2r

0

xBIk

then

n

a contradiction.

n

Since

(h):

=+

= (0).

k=1

+ IO),

I k = J,

kSr (9)

fl I k

So

we a r e t h r o u g h .

trivial.

...,

( h ) -> ( b ) : L e t 1 1 , Inb e m i n i m a l w . r . t . the p r o p e r t y t h a t t h e i r i n t e r s e c t i o n = {Ol, Then e a c h Jr: = Ik (01. Since rsil, n l : J, I,,

+

n

v

+

...,

k4-r b u t J r n I r = f O ? , we h a r e N = Jr;Ir. Hence -9 and J , i s s i h p l e . Jr N/Ir K r i = I1n nIr L e t f o r r E f 1 , ...,n )

-

N

We c l a i m t h a t

= J1;

.

...

...tJr;Kr

and p r o v e t h i s by

i n d u c t i o n on r.

If

r

= 1

then

K,

= I1

and

Assume t h a t i t i s shown f o r the assertion for Since

IrtltKr

Since

YIrt1

= N

and

Jrtl

Also Hence

5

Krtl

Kr

Krtl~Jrtl

9

r (< n).

We show

(by m a x i m a l i t y ) ,

t h e same a p p l i e s t o

i s a maximal i d e a l i n

J r + l $ Kr+l a n d N = J1;

but

= K,

. . . +.J r +.J r t 1 ~ K r t 1

= J1+.

= N.

rtl.

i s simple,

KdKrt1

Jl;If

.

.

K,

...t.J r +.K r

=

,

2c Decomposition theorems

2 . 5 1 R E M P R K S T h e p r o o f o f ( h ) ->

59

( b ) i n 2.50 c o u l d a l s o be done

b y u s i n g s u b d i r e c t p r o d u c t s and "words g e n e r a t i n g p r i m e i d e a l s " s i m i l a r tofMcCoy),

p. 59.

Cf.

also (Higqins),

A t a f i r s t g l a n c e o n e m i g h t assume t h a t " f . g : '

$9.

implies

a1ready"completely reducible'.' T h i s i s n o t t h e case: t h e zero-nr.

N o n t h e d i h e d r a l g r o u p D8 o n 8 e l e m e n t s .

D8 i s k n o w n

Then normal subgroups and i d e a l s c o i n c i d e . B u t G 9

t o have

take

D8

H 4 G,

and

but

H $I

D8

.

By 2 . 4 B ( e )

and 2.12 N c a n n o t be c o m p l e t e l y r e d u c i b l e . 2.52

COROLLARIES ( a ) I f N f u l f i l l s one (and hence a l l ) o f t h e c o n d i t i o n s i n 2.50

I 9 N

and i f

2.48(f),

2.48(e)

t h e n t h e same a p p l i e s t o I ( u s e

and 2 . 3 5 ( b ) ) .

( b ) I f N h a s t h e DCCI a n d i s a s u b d i r e c t p r o d u c t o f s i m p l e n e a r - r i n g s Ni

N

Aqain,

=

8 N jEJ j

(ieI)

then

(apply 2.50(h)

IJcI, J finite: and 1.58).

corresponding statements h o l d f o r N-qroups w i t h

2 . 5 3 D E F I N I T I O N Two d e c o m p o s i t i o n s o f N : N = a r e c a l l e d isomorphic if and

JB's

are

-

/A1 = IBI

-

up t o o r d e r

The K r u l l - S c h m i d t - T h e o r e m

1'1,

acA and t h e

=

N

=

No

I'JB

B€B

Is's

isomorphic.

r e a d s as

2 . 5 4 THEOREM ( R o t h ( 1 ) ) . I f N (,,,r w i t h Ncr),) f u l f i l l s one (and hence a l l ) o f t h e ' c o n d i t i o n s o f 2.50 t h e n any two

(Nr) i n t o s i m p l e i d e a l s a r e

decompositions of N

is o m o r p h ic Proof

.

( f o r rrr.ls)

If

(Iky JL simple)

N =

then

~ ~ t . n, =. tJ ~~; . . . ~ J , N=I1;...;In-l=...=I1={o]

and

N = J ~ ; , . . + J ~ _ ~ = . . . = J ~ ~ ~a or e? t w o i n v a r i a n t s e q u e n c e s with simple factors

60

8 2 IDEAL THEORY

By 2.40

t h e s e sequences and t h e r e f o r e t h e s e de-

c o m p o s i t i o n s a r e isomorphi,c. Compare t h e f o l l o w i n g r e s u l t w i t h 59 o f ( H i g g i n s ) .

2 . 5 5 THEOREM n

( a ) If N =

!'Ir

a n d if I A N

( a l l Ir simple)

r=l

there i s a subset S o f

...,n l

{l,

then

with

(b) I f I,J 9 N a r e such t h a t N / I a n d N/J a r e c o m p l e t e l y r e d u c i b l e t h e n N/In i s completely reducible, too,

a l l o f whose s i m p l e summands b e i n g i s o m o r p h i c t o o n e o f t h e s i m p l e components o f Again,

N/I

or

N/J.

t h e c o r r e s p o n d i n g theorem h o l d s f o r N-groups w i t h

N = No. Proof.

(a) Let

Then

K,

Kr:

...+ ( l s r s n ) , ...,n - 1 1 : Kr 9 N

= ItI1+ I r

= N.

rE{l,

K r n'r+1g1r+1* T h u s we h a v e e i t h e r

or

A

Kr+l

= K,

= Kr'Ir+1.

Kr+l

Hence

KO: = I .

3

T5{1,

...,n l :

N = 1;

l'It,

a n d s o by 2.8

tET

S: = {l, ...,n l \ T

.

( b ) L e t K: = I + J . Then K/I 9 N / I a n d 3 MsN: N / I = = (K/I)YM/l), whence K+M = N a n d K n M = I . So M n J = M n K n J = I n J a n d M+J?I+J = K, M + J 3 M t K = N, h e n c e MtJ = N. Consequently

61

2d Prime ideals

M I I n J = M/MnJ

%

"In

J

N/I nJ

%

MtJIJ = N/J

and

are completely reducible, s o

K / I 9 N/I

is completely reducible by 2 . 4 9 .

The rest follows from (a) and the first line on this page. 2

Ferrero-Cotti showed that N = I , I ? , N # N,, I , * #to}# I 2 implies that all ideals o f N are given by { O l , I , , I 2 and N.

d ) PRIME IDEALS

1 . ) PRODUCTS OF SUBSETS. 2.56

NOTATION If S , T c N then ST: = (StlsESAtET?. For nsIN, the definition o f Sn is then clear.

2.57 PROPOSITION (Maxson ( 1 ) ) . (a)

1 R,S,T

E

N : (RS)T = R(ST).

m

(b) If h: N * and tl SITc

(c)

I 4N

then tl S,T E N : h(ST) = h(S)h(T) W: h-'(YT) ? h-'(?)h-'(T).

VS,T

E

N:

( S t I ) ( T t I ) = STtI.

Proof. (a) and (b) are immediate. (c) follows from (b) for 2 . 5 8 REMARK Note that

T ~ : N* N / I .

S T has no particular structure fn general. Even if S I T are ideals, S T i s not even except i n some very special a subsemigroup o f ( N , t ) cases.

62

$ 2 IDEAL THEORY

2 . ) PRIME IDEALS 2.59 DEFINITION P A N -> I e P v J c P .

i s c a l l e d prime i f

2.60 N O T A T I O N F o r ScN,

let

(Cn}) =: ( n )

I,J4N:

IJCP ->

be t h e i d e a l g e n e r a t e d by S .

(S)

.

2 . 6 1 P R O P O S I T I O N ( V a n d e r Walt ( 1 ) ) . Let P be a n i d e a l o f N . Equivalent a r e

( a ) P i s a prime i d e a l . (b)

v

(c)

v (e) v (d)

I,J A N:

(IJ)

i,jEN: i

4

P

A j

4

1,J A N:

I

a

P

J

I , J A N: i

P

F

I\

P A J

P r o o f , ( a ) ( b ) ( a ) ==> ( c ) : I f s o iEP v j E P .

( c ) -> Then

I s P v J

=+

P.

( i ) ( j ) $ P.

P ->

4

5

4

P ->

IJ

P

IJ $ P.

-7

P.

(e) i s trivial. (i)(j)cP

then

(d): If IZPAJaP, t a k e ( i ) ( j ) $ P , so IJqP.

(i)cP

or

iEI\P

and

(j)cP,

jEJ\P.

( d ) => ( e ) : I f I+PAJ$P, t a k e i E I \ P and jcJ\P. Then ( i ) + P = P a n d ( j ) + P = P . Then ( ( i ) + P ) ( ( j ) + P ) $ P . So 3 i ' E ( i ) 3 j ' E ( j ) 3 p , p ' ~ P : (i'+p)(j'+p')&P. T h e r e f o r e i'(j'+p')-i'j'+i'j'+p(j'+p')&P. But since i ' ( j ' + p ' ) - i ' j k P and p ( j ' t p ' ) E P , i ' j 'BP, hence I J$P

.

2.62 PROPOSITION L e t ( P a ) a E A be a f a m i l y o f prime i d e a l s , t o t a l l y o r d e r e d by i n c l u s i o n . Then P =:P i s a pr ime aEA a i d e a l , too.

n

P r o o f . We may a s s u m e t h a t A is o r d e r e d such t h a t f o r a i B -> Pa c P B . a,B€A

63

2d Prime ideals

Of

P i s an i d e a l . L e t I , J b e i d e a l s o f N . P a -> b a E A : I J 5 P a . I f 3 a c A : I 4 P a '.

course,

fl

IJ 5

acA

then then SO

tl

J E Pa. B l a : J 5 P B , I f 3 y < a : J % Py I c P Y' s o I t P a' a c o n t r a d i c t i o n . acA: J c pa and J c pa.

n

acA

2 . 6 3 PROPOSITION ( M a x s o n ( 1 ) ) . I f I A N i s a d i r e c t summand and P A N i s prime then P n I i s a prime i d e a l i n I. I f JlJ2 F PI\ I (J1,J2 4 I ) then J1J2 c P and J1,J.p N , so J1 c P o r J 2 e P and t h e r e f o r e J1 c P n I o r J 2 c P n I .

Proof.

2.64 PROPIISITION I f I 3 N a n d I c P A N a n d i f a : N -+ N / I = : N i s t h e c a n o n i c a l e p i m o r p h i s m a s u s u a l t h e n : P i s p r i m e c-> n ( P ) i s p r i m e . J ~ J ~ ~ ~ (~ P l) ;Aj IT), ~ l e t J ~ : =a - l ( ~ ~ ) -1 -1 ( i ~ { 1 , 2 1 ) . By 2 . 5 7 , J 1 J 2 = TI ( J l ) r ( J p )c If

P r o o f . ->:

ra-1(JlJ2)En-l(Tr(P)) = P t I = P. So or

JlcP v J2cP, hence J2cn(P).

( b ) = > ( c ) a n d by 1 . 5 8 a n d 2 . 9 1 , we o n l y h a v e t o show => i n ( a ) . S u p p o s e p s T ( I ) \ I . S i n c e N\I i s a n s p - s y s t e m , 2 . 9 2 p r o v i d e s u s w i t h a n m-system M s u c h t h a t

___ Proof.

aEMsN\I.

But

M A I C ( N \ I ) ~ I= @ c o n t r a d i c t s

asrfI).

F o r more a n d some o t h e r r e l a t e d m a t e r i a l s e e B e i d l e m a n ( 7 ) . F e r r e r o - C o t t i ( 7 ) , G o j a n ( l ) , Oswald ( 5 ) , ( 8 ) and R a m a k o t a i a h Rao ( 5 ) , S a n t h a k u m a r i ( 2 ) . S e m i p r i m a r y n e a r - r i n g s w e r e c o n s i d e r e d in a s e r i e s o f papers by K a a r l i . See in p a r t i c u l a r K a a r l i ( 7 ) and 9.260.

2e Nil and nilpotent

69

e) NIL A N D NILPOTENT

2.96

DEFINITION

(b)

S

N

C

is called nilpotent if is called nil i f all

(c) S e N 2.97

3

i s c a l l e d n i l p o t e n t if

(a) nEN

k E I N : n k = 0.

3

kcIN: S k = CO)

SES

are nilpotent.

REMARKS ( a ) S C N n i l p o t e n t - > S nil. ( I n 3 . 4 0 w e w i l l s e e t h a t i f NEW has the DCCN t h e n " n i 1 " a n d " n i l p o t e n t " c o i n c i d e f o r N-subgroups.)

T

(b) S 5

2.98

2.99

S

N

A

nil(poter,t) => S nil(potent).

T

EXAMPLES ( a ) In

n4[xJ,

( b ) If

ncNc

COROLLARY If

2x

i s nilpotent.

is n i l p o t e n t t h e n

I A N

n = 0.

i s nil t h e n

I

c

No.

P r o o f . B y 2.18, I = IotI C' so by 2.97(b) Ic = NcnI i s n i l , h e n c e by 2.98(b) I c = ( 0 1 a n d I = 1, c N 0' 2.100 T H E O R E M ( R a m a k o t a i a h ( 3 ) ) . I 9 N . I a n d N/I a r e nil(potent).

N i s n i l ( p o t e n t )

Proof (for nilpotence) ==>: by 2.97(b), I is n i l p o t e n t . I f 3 k c l N : N k = CO} t h e n (N/I)k kl'Nk: k2€IN : I

b

NY = ?!

(a):by

3.4(d)

Proof.

Nr

o f t y p e 0.

R = {oj

N = No-group

i s o f type 2.

ycr* :

->

Nr

then

r.

R =

or

is of

I n t h i s case,

(see a l s o 3.19(a)!). and 1.34.

( b ) f o l l o w s f r o m ( a ) and 3 . 2 . ( c ) ->:

Let

Nr

~ E A : N6 = { o )

s i n c e each

M'm; t M'(mi) 2 t 3 kEIN : M'(mi) k = M'(m;) ktl = . Thus (M'(mi)k)(mi)kt' = M'(m;)k and s i n c e m i : = = ( m i ) kt 1 E M ' ( m i ) k , m i g e n e r a t e s M'(mi) k . mi

...

By t h e minimality o f

M',

(0:m;)n

...

=

{Ol.

A g a i n using the minimality o f M I w e see that e a c h h a s ( 0 : m i ) n M'mj = g e n e r a t o r m i o f M'(mi)k=W'm; = (0).

We shall s h o w t h a t

mi: = mimi violates this statement. (a) m i m j generates M ' m j , for mi = m j m i and M ' m i = M ' imply t h a t (M'mj)(mimi) = = M'mimj = M ' m i . f C O I , f o r otherwise M ' -N M ' m j 5 M'm'2 < MI. T a k e s o m e n o n - z e r o mgc(0:mj). 3 mieM': mkmi m i , since m i generates M ' . 6 3 1 3 ' Now 0 = m;mj = m k m i m j = m'm'm'm' H e n c e mkmje(0:rnimj)n M ' m j , b u t m k m i $. 0 s i n c e mimimi = m k m i = m k 0.

( b ) Observe that

(0:mi)

%

+

S o we a r r i v e a t

a c o n t r a d i c t i o n and the p r o o f is

complete. N-aroups o f type 0 o v e r a semiprimary ( s e e 9 . 2 6 0 ) near-ring N are studied in Kaarli ( 2 ) , ( 4 ) and (6). Holcombe-Walker ( 1 ) study N-aroups Nr o f type 3 (i.e. ,.,r is o f type 2 with (f/ n E N : ny = ny')=> y = y ' . The s u m of all left ideals L o f N = N o , where NL i s o f type 1 , i s called socle o f N (see e.a. Ramakotaiah (3)).

81

3b Change of the near-ring

b ) C H A N G E OF T H E N E A R - R I N G

Up t o now we h a d a n u n j u s t s i t u a t i o n : a n e a r - r i n g

keeps an harem o f N - g r o u p s , b u t n o t c o n v e r s e l y . Now we l e t an N - g r o u p N r c h a n g e i n t o N / I r ( f o r some I A N), N o r , N cr . T h e s e changes w i l l be an i m p o r t a n t t o o l i n l a t e r c o n s i d e r a t i o n s . 3 . 1 4 PROPOSITION ( B e t s c h ( 3 ) ) . L e t I b e a n i d e a l o f N , group and u E I 0 , 1 , 2 1 .

(a) If

r

i s a n N-group w i t h

(o:r)

I c

r a

then

( n t 1 ) y : = ny

r

makes

i n t o an

N/I-group

N/Ir.

I f Nr i s o f t y p e w , s o i s If (b) If

Nr

N/Ir* i s f a i t h f u l , t h e same a p p l i e s t o

N/Ir'

r i s a n N/I-group t h e n ny: = ( n t I ) y

makes If If

r i n t o an N - g r o u p

N/Ir i s NIIr i s

Nr

with

o f t p y e v , so i s f a i t h f u l then

I

C

(o:r)"

Nr.

I = (o:r)N.

The p r o o f i s a c o l l e c t i o n o f s t r a i g h t f o r w a r d a r g u m e n t s a n d therefore omitted. Observe t h a t ( N / I ) o = { n o t I l n o c N o ) . and a s an Each N - g r o u p r c a n b e v i e w e d a s an N o - g r o u p N,-group r i n a n o b v i o u s way ( b y r e s t r i c t i o n ) . I n 3 . 4 ( d ) NC

we a l r e a d y m e n t i o n e d t h i s f a c t . We now s t u d y t h e r e l a t i o n between r, r a n d Nr: NO

NC

8 3 ELEMENTSOF THE STRUCTURE THEORY

82

r be a n N - g r o u p and A a s u b s e t o f r . ( a ) Nr is faithful i f f r and r a r e faithful.

3.15 P R O P O S I T I O N Let

NC

NO

( b ) A AN

r

A AN,

( c ) A sN r A

r r

5

A R c A.

NO

Proof. ( a ) I f Nr i s f a i t h f u l , t h e s a m e t r i v i a l l y a p p l i e s r and r . C o n v e r s e l y , l e t n r b e = {03. to NC

NO

T h e n ( w i t h n = no+n C a s in 1.13) \I YEr: noy+nco = = n o y + n c y = ny = 0 . T a k i n g y = o y i e l d s nco = 0 . So y c r : noy = o and no = 0. But nco = o g i v e s y E r : n,y = 0, h e n c e n c = 0. T h e r e f o r e n = n o +n c = 0. ( b ) => is t r i v i a l . If

A AN

r

then

0

i

6cA yEr ncN: n(6ty)-ny = no(6+y)tnc(6+y)- n c o - n 0y = n 0 (6ty)+nco-n,o-noyEA. ( c ) is even m o r e trivial.

Nr a n d

The r e l a t i o n b e t w e e n 3.16 COROLLARY

Let

Nr E

NO

r

i s particularly important.

N(ZI. r

( a ) Nr is s i m p l e NO

is simple.

(b) Nor is m o n o g e n i c by y ->

Nr is m o n o g e n i c by

( c ) Nor is s t r o n q l y m o n o g e n i c -> genic o r (d)

iol

r is N,-sirnple

!,r

y.

is s t r o n n l y m o n o -

+ R + r. ->

r is N-simple.

3.17 EXAMPLES I f N = Mc(Z4) t h e n Z4 is N - s i m p l e but not No-simple ( s i n c e IO,23 i s a n No = IO3-subgroup). So N - s i m p l i c i t y d o e s n o t i m p l y N,-simplicity.

P l u g g i n g all t o g e t h e r y i e l d s

83

3b Change of the near-ring

Nr

3.18 THEOREM Let

be an N - g r o u p and

( a ) Nr i s of t y p e w

r

-?

wc{O,l,21.

Nor = C o ) .

i s of type w o r

NO

( b ) Nor i s o f t y p e w ( f o r

Nr n

or

= Co)

Nr

P r o o f . ( a ) Anyhow,

r)

=

Q.

w = 1

assume t h a t i n

Nr i s o f t y p e

->

w.

r

i s simple, t h e r e f o r e a l s o NO

by 3 . 1 6 ( a ) . L e t Nr be monogenic by y . Then Hence Noy = ( 0 1 o r Noy = r . If

Noy =

r,

r

Noy

sk

r

by 3 . 4 ( a ) .

i s monogenic, t o o .

NO

If

Noy =

(01

then

r

= Ny = NoytR = R

implies

t h a t b YEr: Ny = r . Again by 3 . 4 ( a ) , b y e r : Noy = C o ) o r = r . So e i t h e r r i s monogenic o r N o r = C o ) . If

Nr If

NO

i s of t y p e 1 t h e n

{ol

or

R =

r.

t h e n b y c r : Ny = NoytR = Noy i s a g a i n o f t y p e 1.

C2 =

r

=

Col

and

l\10

If = r t h e n e a c h y e r g e n e r a t e s Nr s o ( a g a i n by 3 . 4 ( a ) ) t/ Y E r : Noy = I01 o r Noy = So r i s e i t h e r o f t y p e 1 o r Nor = Io).

r.

NO

The a s s e r t i o n f o r

w = 2

i s trivial.

( b ) By 3.16.

3.19 R E M A R K S ( a ) 3 . 1 8 ( a ) and ( b ) show t h a t 3 . 7 ( c ) h o l d s f o r a r b i t r a r y near-rings! ( b ) Information a b o u t t h e behaviour of

can be found i n M l i t z ( 3 ) .

Mr

with

M sN N

$ 3 ELEMENTSOF THE STRUCTURE THEORY

84

c ) MODULAR1 TY

3.20

L A,

DEFINITION

3

b

eEN

N

i s c a l l e d m o d u l a r :

n&N : n - n e c L .

I n t h i s c a s e we a l s o s a y t h a t L i s " m o d u l a r b y e " and t h a t e i s a " r i g h t i d e n t i t y modulo L " ( s i n c e

b

ncN:

ne

n (mod L ) ) .

f

3 . 2 1 REMARKS

(a) If

A,

L1,L2

L1

with

N

t h e n L 2 i s m o d u l a r b y e, ( b ) I01

L2

5

i s modular by e

a n d L1

too.

i s modular i f f N contains a r i g h t i d e n t i t y .

( c ) E v e r y normal subgroup o f (Nc,+) i s a modular l e f t i d e a l o f Nc {by a n y e l e m e n t o f N c ) ,

(d) I f L i s modular by e i n

Nc'YlO

then

ecL

iff

3 . 2 2 PROPOSITION ( B e t s c h ( 3 ) ) . E a c h m o d u l a r l e f t i d e a l i s c o n t a i n e d i n a maximal one ( w h i c h i s modular, Proof.

L =

N.

L)NEV, too).

L e t L b e m o d u l a r b y e . A p p l y Z o r n ' s Lemma t o t h e

I

set o f a l l l e f t ideals

3

L

with

e

4

I

and

use 3.21(a). P r o p o s i t i o n 3.22 3.23

i s n o t a l w a y s t r u e if N

PROPOSITION ( B e t s c h ( 3 ) ) .

3

re 9 N N

P r o o f . ->:

3

ycr:

Nr

L 3, N

f

= {n+LlncN)

y:

i s m o d u l a r

n ( e t L ) = L

L

2

(L:N).

r

P r o o f , T a k e some ( b y y ) m o n o q e n i c N - g r o u p L = (o:y)

Then 3.22

-

(o:r)

3

= (o:N/L)

with

3.24 a r e s i m i l a r t o t h e r i n g case ( ( J a c o b s o n ) ,

Looking a t

/L:N)

L =

( 0 : ~ ) .

= (L:N).

more c l o s e l y g i v e s f o r f u t u r e use

pp.

5-6).

(cf.

Ramakotaiah ( 1 ) ) :

3.25

L e t L b e m o d u l a r b y e. Then

PROPOSITION

(L:N)

= (L:Ne)

and t h i s i s t h e g r e a t e s t i d e a l o f N c o n t a i n e d i n L . Proof.

b b

c (L:Ne)

(L:N)

i s clear.

n'EN : nn'eEL. n'EN:

nn'EL.

But

If

n&(L:N)

So

nE(L:Ne)

nn'-nn'eEL, and

then

hence (L:N)

= (L:Ne).

By 1 . 4 2 , (L:N) i s a l e f t i d e a l and i t i s e a s y t o s e e t h a t i t i s e v e n a n i d e a l o f N , (L:N) 5 L h o l d s t y 3.24.

If

I 9N

I

with

5

L

then t r i v i a l l y

1

C

(L:N).

3 . 2 6 THEOREM ( c f . ( K e r t e s z ) , p , 1 2 2 ) . I f N = Ll+L2, where Ll,L2 a r e modular l e f t i d e a l s , then L 1 n L 2 i s again ~

modular. Proof. Let

L1,L2

Decompose

be modular b y

el,e2,

el = %+%2 e 2 = "1+%2

where

We c l a l m t h a t

L1n L2

If

n-ne = n-n(k21+t12)

-n( But

ntzN

then

kll,f.21~Ll,

n-nelEL1,

(51++)

Therefore Similarly,

b

k12'k22EL2'

i s modular by

%21+k12) = n-nel+nel-n

nk12-n

respectively.

el,e2:

't21+k112 = : e .

= n-na12+nt12-

( - E l l+el ) + n t 1 2 - n ( R Z 1 + L l 2 ) .

nel-n(-kll+el)EL1

and

'L1ntzN:

n-necL1.

nEN: n - n e E L 2 ,

a n d we a r e t h r o u g h .

86

$ 3 ELEMENTS OF THE STRUCTURE THEORY

3.27 C O R O L L A R Y

( a ) I f L i s a m o d u l a r a n d M a maximal m o d u l a r l e f t i d e a l then L n M i s modular. ( b ) A f i n i t e i n t e r s e c t i o n o f maximal m o d u l a r l e f t i d e a l s

i s modular.

( t ) I f N i s a d i r e c t sum o f two m o d u l a r l e f t i d e a l s t h e n N contains a riqht identity. ( d ) ( B e t s c h ( 3 ) ) . I f N c o n t a i n s a f i n i t e f a m i l y o f maximal modular l e f t i d e a l s w i t h z e r o i n t e r s e c t i o n t h e n N contains a right identity.

D-E F I N I T I O N L e t v be ~ ~ 0 ~ 1A l ~e f t2 i d~e a .l L o f N i s 3.28 c a l l e d v-modular i f L i s modular a n d N / L i s an N-group ( v i a n ( n ' t L ) : = n n ' t L ) of type v. L e t gv(N) be t h e s e t o f a l l u - m o d u l a r l e f t i d e a l s o f N . 3.29 R E M A R K So a 0-modular l e f t i d e a l i s j u s t a modular m a x i m a l o n e and a 2 - m o d u l a r l e f t i d e a l L i s a m o d u l a r maximal l e f t i d e a l w i t h no N o - s u b g r o u p s t r i c t l y b e t w e e n L a n d N . (Beidleman c a l l s t h e s e l e f t i d e a l s " s t r i c t l y m a x i r a l " . ) v - m o d u l a r l e f t i d e a l s t u r n o u t t o be v e r y u s e f u l i n d e t e r m i n i n g r a d i c a l s of r e l a t e d near-rinqs. 3 . 3 0 PROPOSIlION L e t p r o d u c t . Let L i Denote

ll M j jEI

( N i ) i c I be n e a r - r i n g s a n d N t h e i r d i r e c t be a l e f t i d e a l o f N i f o r some i E I .

with

Mj:

Then f o r v ~ C 0 , 1 , 2 ) , L i v-modular i n N .

[

=

i + j Nj Li

by

i = j

i s v-modular i n N i

ITi qk N . iff

Li

is

Proof. ( a ) I f Li i s v-modular i n N i then (Ni/Li) n i ( ( . . . , n: , . . . ) t L i ) : Ni = i s o f type v. By = (... , O , n . n ' , O , , . . ) t r i , becomes a n N i - g r o u D 1 i 2, a n d c l e a r l y N/Ci = Ni/Li. S o N/Ti i s an

N/ri

Ni

3c Modularity

If

o f t y p e v.

Ni-group

the statement),

Eli ( n o t a t i o n

Ji:=

% =

Ni

a7

s o 3.14(b)

N/Ji,

as i n

shows t h a t

i s an N-group o f t y p e v (and t h e m u l t i p l i c a t i o n N/Ei i s t h e same a s i n 3 . 2 8 ) . H e n c e Ei i s v - m o d u l a r i n N. (b)

ri

If

N/Ti

i s v-modular i n N then

N-group o f t y p e v. S i m i l a r t o ( a ) , (...,ni,...)(n;tLi):=

where

of Ni/Li 3.14(a) of 3.28. From 2.28, 3.31

2.30

N

i n Ni/Li

Li

=N Ni/Li

The a n n i h i l a t o r

(as i n (a)),

i s an Ni-group

Therefore

N/SCi

n 1. n !1t L i .

Ji

contains

i s an 'Ir

s o by

of type v i n t h e sense

in

i s u-modular

Ni.

a n d 3 . 3 0 we g e t

I e- N

COROLLARY I f

i s a d i r e c t summand i n N a n d i f L c y v ( I )

t h e n t h e r e i s some L t Y , , ( N ) T h e r e e x i s t e x a m p l e s ( s e e e.g. o f l o w o r d e r ) such t h a t 3.31

L = CnN.

with

N) 1 ) i n t h e a p p e n d i x ( n r .

no.

' S

does n o t n e c e s s a r i l y h o l d f o r v = l i f

I i s n o t a d i r e c t summand. See a l s o E x . 6 . 3 2 i n Y e l d r u m ( 1 3 ) . Is,

i n 3.30,

i s a counterexample 3.32

ci?

e v e r y L c Y V ( N ) g i v e n b y some

L

=

i(x,y)(x:y

mod 2)

i n t h e c o n s t a n t n e a r - r i n g on k x 2 . B u t :

P R O P O S I T I O N L e t N b e t h e d i r e c t sum ( o r p r o d u c t ) o f t h e n r . ' s Ni

EL!.

iI ) ~ and

(

I f QNi$L

vided t h a t Proof.

L 2,N.

For i

E

I

E ~ ~ ( tNh e)n L i c

N = N0 i f

l e t L i : = { l . €1 N . 11 (

z,(Ni)

If L i s m o d u l a r b y ( . . . , e i , . . . ) I f a l l L .1= N .1 t h e n

t3N.G 1

L.

I. ( 0 ) I f v = O , assume t h a t Li i n Ni.

t h e n Li

i s m o d u l a r b y ei.

S o s u p p o s e now t h a t Li

Ni

4

i s n o t a maximal l e f t i d e a l strictly

Now i f L~:={(..,Oy1~,O,..)~l!~L~}

and Ni.

L+C; i s

1

a l e f t ideal i f N properly containing =

N.

T a k e n .1€ N .1\ L ! .1

(..,li,..)t(..,O,l;,O,..)

a l l j+i a n d n i = l i + l ; So Li

i s a l e f t i d e a l o f Ni.

T h e n t h e r e i s some l e f t i d e a l L;

L , w h e n c e L t L-; =

I (pro-

E

b e t w e e n Li then

E

v=I).

I t i s e a s y t o show t h a t e a c h Li

f o r some i

. . . ,O , l i , O , . . . )

f o r some i

E

E

Li+L;=L;,

L+L!. -1

T h e n (..,O,ni

,O,..)

Hence 1 . = O f o r J

a contradiction.

i s a 0 - m o d u l a r l e f t i d e a l i n Ni.

88

8 3 ELEMENTS OF THE STRUCTURE THEORY (2)

I f w=2, Li

we p r o c e e d s i m i l a r l y .

i s a maximal l e f t i d e a l .

L i s 0-modular,

too,

m a x i m a l t h e n t h e r e i s some b i g g e r ( N i ) o - s u b g r o u p Define L;

s t r i c t l y monogenic. L . < N.(n.+Li) 1

We m u s t show t h z t N i / L i

S u p p o s e t h a t ni€Ni

< Ni/Li.

1

If n:=(.

N(ntL)=O+L o r N(n+L)=N/L. =

( . . ,O,Nini,O,..) 1

n"=(.

t a k e some n;ENi

w i t h n'+L=n"n+L.

we g e t n 1! - n '1! n1. ~ L1 ., w h e n c e n 1! + L i contradiction. 3 . 3 3 COROLLARY

t h e n Ti 3.34 -__

If N

H e n c e Li

=ONi

i s in N,

SNSS

=NO

t h e n L i s an i d e a l o f NS and S/L If n E N ,

Nn =

a contradiction. not i n

w i t h n;+Li

T h e r e i s some

I n t h e i - t h component E

Ni(ni+Li),

i s I-modular

( a n d N=No f o r

(as d e f i n e d i n 3.30)

THEOREM ( K a a r l i ( 4 ) ) I f S
M contains a right identity e with Ne = Me = M (see also Beidleman (6)). ( b ) If

L 4, N

is a minimal non-nilpotent left ideal then L contains a non-zero idempotent.

(c) (Beidleman (1)). I f L A, N is a minimal non-nilpotent N-subgroup then L is a direct summand o f ". Proof. (a) If mEM is not nilpotent then m 2 EMmEM is M and by 3.13 M contains not nilpotent, s o Mm a r i g h t identity e. By t h e minimality o f M, N e (not nilpotent!) = M = Me. (b) By t h e minimum condition in N , L c o n t a i n s a minimal non-nilpotent N-subgrour, M. M has a right identity e by (a) and s o L has a non-zero idempotent e. (c) L contains a riqht identity e by (a) with L e = L . and L i s a direct summand o f ,,,N.

By 1.13, N=L+(r):e)

96

3 3 ELEMENTS OF THE STRUCTURE THEORY NEW,,

3 . 5 2 COROLLARY ( B l a c k e t t ( 2 ) ) .

DCCN,

N w i t h o u t non-zero

M

n i l p o t e n t N - s u b g r o u p s . Then e a c h m i n i m a l N - s u b g r o u p

is

g e n e r a t e d b y an i d e m p o t e n t e w h i c h i s a r i g h t i d e n t i t y o f M. We now t u r n t o m i n i m a l i d e a l s .

3.53

PROPOSITION ( S c o t t

M nilpotent,

L e t I b e a m i n i m a l i d e a l , rl

(4)).

M c I.

sFT N ,

I M = (0).

Then

Mk be = t o ) , k 2 2 , M k - I p {O). T h e n M ~ - ~ . M= {OI, s o M k - l i s contained i n the ideal Mk-1 i s (0:M). Hence t h e i d e a l J g e n e r a t e d b y c o n t a i n e d i n (0:M). S i n c e (0) J E I, J = I a n d

Proof. Let

I M = {Ol. See a l s o K a a r l i

( 2 ) and Scott

(16)

NE~,,

3.54 THEOREM ( S c o t t ( 6 ) ) .

DCCN,

I a minimal i d e a l .

Then I i s a f i n i t e d i r e c t sum o f N - i s o m o r p h i c m i n i m a l l e f t ideals o f

N

(and t h e r e f o r e c o m p l e t e l y r e d u c i b l e

NI).

when c o n s i d e r e d a s

-Proof.

We n e e d 3 l e m m a t a a n d k e e p t h e a s s u m p t i o n s o f t h e

theorem.

Nr b e f a i t h f u l a n d

Lemma 1. L e t

Nr.

ideal of

{Of

Let

A be a minimal

L c (A:r)

be a

r

v Ly = {oI. T h e n L i s a f i n i t e d i r e c t sum o f N - i s o m o r p h i c m i n i m a l l e f t i d e a l s o f N. l e f t i d e a l such t h a t

Proof.

3 If

Nr

(a) Since

L$(O:Y~),

YIEr: (

0

“(o:Y1)L

:

PI

~

~

){Ol ~

yEr:

Ny =

i s f a i t h f u l and

so then

L

$. {Ol,

( 0 : ~ ~= )(o:yl)nL=L. ~

3

Y ~ E ~( :0

:

~

= ~

(o:Y*)L.

P r o c e e d i n g i n t h i s m a n n e r , b y t h e D C C N we e v e n t u a l l y o b t a i n elements

yl,y2,.

..,yn€r

)

~

97

3f More on minirnality

Anyhow, we g e t a n o n - e m p t y s u b s e t C o f Cyl,.

. . ,ynl

of minimal o r d e r f o r t h e ( o : C ) ~ = EOI.

property that Set

C = : Cul,

(b) Define

...,o k } .

L1:=

L

We now show t h a t

hi:

=

k>l Li

morphisms.

k

k = 1

if

= (O:C\CU~})~ if

Li:

11

and

(iEC1,

are N-iso-

A

+

...,k l ) .

&ai

-+

101 $. L = L 1 a n d ( e ~ : u ~ ) ~ = C O f . Lol = L l u l Col a n d Nol = r .

1: T h e n Thus Since

LE(A:r),

Llol 9,i Nul

=

LlolcA.

By 3.4(a),

r.

Since A i s minimal,

Llul

and

= A

hl i s surjective. Also, K e r h l = ( 0 : ~ ~= )EOI; ~ hence hl i s an N - i s o m o r p h i s m . k > 1: S u p p o s e t h a t

3

jc11,

...,k l :

L-o ~

-

j L =(o:o.), so jJ L =(o:C), = COI, a contradiction t o jt h e m i n i m a l i t y o f E. = to}.

Then

Hence a l l

Lioi

Lioi

Also,

= A.

=+

Col

and ( a s a b o v e )

K e r hi

= (o:oi)n =

COI,

(0:UCuil)n a n d a g a i n hi

= (o:ui)n

( 0 : ~ )-

L = L i s an N-iso-

m o r p h i sm.

...,

( c ) L e t i be ~ { l , k l . Li i s a m i n i m a l l e f t i d e a l o f N: BY 3 . 4 ( e ) ,

Liui Thus

= A

c\r

N/(o:oi)

-N

=

r.

BY ( b ) ,

i s minimal.

L i + ( o : u i )/(o;oi)

of t h e N-group Li n ( o : o i )

Nui

= Col

i s a minimal i d e a l

N/(o:oi). (by (b)),

-

Since 2.8 gives

Li

=

98

53 ELEMENTSOF THE STRUCTURE THEORY

(d) Since a l l isomorphic.

Li

% aN

A,

t h e L i ' s a r e N-

...

tLk. We may ( e ) We show t h a t L = L 1 t assume t h a t k > l . I f R c L , V i c { l , ...,k l : RuicA. By ( b ) , L .1u . 1 = A , S O 1 R i € L i : Rui = Riai.

S e t El:= R 1 t , . . t R k . I f i += j, ~ ? . c L ~ ~ ( o : us ~o ) , 2 . 0 0. J J i So f o r a l l i c { l , k l Riai = R i u i . T h e r e f o r e (I?-J!')E = (01, so R-R'c(o:E)hL= = {I-)}, Hence R = I?' = e l + ...+R k .

...,

( f ) The sum i n ( e ) i s d i r e c t : 'Pk I f 11 = R 1 t . . . + R k = P I +

...

Then

V

iE{l,

...,k l :

Lai = piui

( R i, p i E L i ) = Riui.

...,

Thus ic{l, kl: R i - p i ~ ( o : C ) nL t h e proof i s complete.

and

Lemma 2 . I f I N {Ol a n d M SN N i s m i n i m a l f o r I M =/= ( 0 1 then (a) M contains a right identity. (b)

In M

i s minimal amongst a l l n o n - z e r o i d e a l s

of the nr. M which a r e a l s o N-subsroups.

(c) In 4 I

i s t h e sum o f m i n i m a l i d e a l s o f

NM.

P r o o f . ( a ) We s h a l l show t h a t MM i s m o n o g e n i c . I f kf m e M : Mm 3. M, I M M = ( 0 ) s i n c e V m c M : I ( M m ) = {Ol b y t h e m i n i m a l i t y o f M . D e n o t e t h e i d e a l g e n e r a t e d b y It4 b y J. S i n c e It4 i s c o n t a i n e d i n t h e i d e a l (O:M), Js(0:M) a n d J M = 101. But I M (01, I M c I a n d so J = I a n d we a r r i v e a t t h e c o n t r a d i c t i o n I M = (01.

a

99

3f More on rninimality

So M i s m o n o g e n i c a n d t h e r e s u l t ( a ) f o l l o w s f r o m 3.13.

I M p I01 and IMcInM, I n M i s a n o n - z e r o i d e a l o f M. L e t I01 K 9 M b e s u c h t h a t K sN N and K c I n M . S i n c e KMEKEI, Kc(K:M)n I a n d ( K : M ) n I p {OI, s o ( K : M ) n I = I a n d Ic-(K:M). I t f o l l o w s t h a t IM'(K:M)MGK. By ( a ) , M c o n t a i n s a r i g h t i d e n t i t y e , s o ( I n M ) e = I n M. Thus I n M!(! I n M ) M E I M c K (b) Since

and ( b ) f o l l o w s .

9 IOI,

In M

(c) Since

minimal i d e a l W o f T a k e some

there exists a

NM

in

I n M .

mEM,

= I01 o r NM ( s i n c e Wm i s an N-endomorphic image o f W ) . I f Mm $. M t h e n I M m = {Ol. But M contains a r i g h t i d e n t i t y e , s o We = W a n d t h u s Wm = WemrIMm = i O l .

Mm = M

If

then

Wrn

i s either

a minimal i d e a l o f

Hence

mcM: Wm =

ideal o f

I01

or

i s a minimal

Wm

NM.

L

W = WeC-L a n d

9

(01.

S o L i s t h e sum o f m i n i m a l i d e a l s o f

L d R M.

O f course,

1 WmiTieL,

=

so

Also

L sl M,

;EM:

NM.

LiTi =

and L i s a non-

mcM

zero ideal o f M contained i n L

5:

In M

I n M.

By ( b ) ,

a n d ( c ) i s shown.

A l t h o u g h t h e r e a d e r m i g h t b e g a s p i n g f o r b r e a t h , we n e e d a t h i r d Lemma, w h i c h w i l l b e u s e d i n t h e p r o o f

o f 4.47.

$ 3 ELEMENTS OF THE STRUCTURE THEORY

100

Lemma 3 .

I N dp I 0 1

If

t h e r e e x i s t s an N-group

and a m i n i m a l i d e a l (a)

Nr

such t h a t

( o : r ) n I = (0).

( b ) I+(o:r)

(c)

r

A SIN

b

YEr:

C

(Azr).

M, W ,

Proof. Let

e

InM

NM.

minimal i d e a l s o f

I nM,

i n lemma 2. B y p a r t

be as

( c ) o f t h i s lemma, W =

r.

IY = ( 0 1 v NY =

i s t h e sum o f

By t b e f a c t t h a t

we c o n c l u d e f r o m 2 . 4 8 ( e ) t h a t W

i s a d i r e c t summand.

3

SO

U

Define

43

In

InM:

r:=

M/U

M = WU ;.

and

‘L

=N W .

InMIU = W;U/,

A:=

So A i s a m i n i m a l i d e a l o f

NI’.

We n o w p r o v e t h e lemma.

(o:r)n I

(a)

= CO1

or

= I.

If

I e ( o : r ) . So I r = { o } a n d ( I n M)e = I A M y s o I n Ms( I But Thus A = ( 0 1 , a c o n t r a d i c t i o n .

(o:r)nI = I IMEU.

then

A

M)McIM~U.

( c ) I f Y E r , 3 mEM: y = m t U . I f Nm=M, the m i n i m a l i t y o f M gives us I ( N r n ) = (0).

So

INe(o:y).

t h e l e f t i d e a l generated by (IN)%5(o:y).

By 1 . 5 2 ,

generated by I N So

I y = Col,

(IN)I1

If

i s

(IN)I1

IN

then i s the ideal

and t h e r e f o r e e q u a l s I . completing t h e proof.

T i r e d , b u t h a p p y we a r e r e a d y f o r t h e P r o o f o f t h e theorem,

Suppose t h a t

I N = COI

and L i s a

m i n i m a l l e f t i d e a l o f N c o n t a i n e d i n I . So and

L A N.

Thus

L = I

LN =

and t h e theorem h o l d s .

COI

101

3f More on minirnality

If

p

IN

N/

(0:

{O),

r1

let

r,A

N ur

=:Nu,

be as in Lemma 3 . If is faithful and has A as a

minimal ideal. Set I & : = I t ( o : r ) / ( o : r ) . lemma 3(c), tl y c r : ( N a y = r v I l y = c o ) ) . By lemma 3(a), I I/(o) 'b 1 ' . Thus by % lemma 1, I -N 1' is the direct sum o f minimal N - i somorphic 1 eft i deal s . By

zN

nN

The p r o o f is now complete. Note that if NcnO i s simple and has the DCCN then N N is completely reducible and 2.50 i s applicable. Cf. 4 . 4 6 and 4 . 4 7 . 3.55 COROLLARY (Scott ( 4 ) , ( 6 ) ) . NEW,, DCCN, I a non-nilpotent minimal ideal o f N; O ( N ) : = sum o f all nilpotent left ideals. Then O ( N ) r l I = ( 0 ) and Q ( N ) is nilpotent. Proof.

See S c o t t ( 1 ) o r ( 4 ) f o r t h e p r o o f t h a t Q ( N ) i s n i l -

potent. of

V

By 3 . 5 4 a n d 2 . 4 8 ,

NI. L e t L iE1

v

2

Q ( N ) n I i s a d i r e c t summand I be s u c h t h a t I = Q ( N ) n I ; L . By 2 . 2 2 .Q.

qeQ(N)n I

t, R E L : i ( q + R ) : i q + i R (mod Q ( N ) n I n L ) .

Since ( Q ( N ) n 1 ) n L = {O},

i ( q + k ) = i q + i R . Hence

I' = I ( Q ( N ) n

I+L)

I ( Q ( N ) ~ I)

COI, since Q(N) i s n i l p o t e n t . so

=

S I(Q(N)nL)+IL.

B u t b y 3.53, I~SILSI

and t h e l e f t i d e a l g e n e r a t e d by 1 2 , t h e i d e a l g e n e r a t e d b y 1'

(by 1.52)

and

I (by minimality) coincide.

So I i s c o n t a i n e d i n t h e l e f t i d e a l g e n e r a t e d b y I L C L , I = L and Q ( N ) n I = { O } . 3.56

REMARK There a l s o e x i s t r e s u l t s c o n c e r n i n g n e a r - r i n g s w i t h

ascending chain conditions. Oswald ( 2 ) . (4),

For "Goldie-type"

ones,

F o r more r e s u l t s , c o n s u l t S c o t t ( l ) ,

D i Sieno- D i Stefan0

and Zand ( 1 ) .

see

Kaarli

(2),

(4), R a m a k o t a i a h - S a n t h a k u m a r i ( 1 )

102

$ 4 PRIMITIVE NEAR-RINGS

This paranraph presents a discussion o f t h e " b u i l d i n q stones, near-rinqs

a r e made o f " ,

Similar t o r i n q theory,

the so-called t h e "atoms"

"primitive near-rinqs".

a r e n o t t h e simple near-

r i n g s as one m i o h t e x p e c t a t a f i r s t g l a n c e . an i m p o r t a n t c o n n e c t i o n (4.47).

g i v e n a n e a r - r i n q N,

f r u i t s (= N-groups)

r e c o q n i z e them b y

we l o o k a t a l l o f i t s

and ask, w h e t h e r t h e r e a r e f a i t h f u l and

' ' e n o u g h s i m p l e " o n e s amonq t h e m . N " p r i m i t i v e on t h i s N-group". p r e c i s e we f i x

however,

The i d e a t o c o n s i d e r p r i m i t i v e

n e a r - r i n g s comes f r o m t h e b i b l e ( " Y o u w i l l their fruits"):

There i s ,

If t h i s i s t h e c a s e , we c a l l

S i n c e "enouqh s i m p l e "

i s not

i t s menninq i n w a n t i n g N-qroups o f t y p e v.

The r e s u l t i n q c o n c e p t i s t h a t o f " v - p r i m i t i v i t y " . We g e t t h e h i e r a r c h y 2 - p r f r n i t i v i t y < T > l - p r i m i t i v i t y < Z ' mitivity,

discuss conditions,

O-pri-

w h i c h f o r c e some o f t h e s e c o n c e p t s

t o c o i n c i d e a n d make a l o t o f w o r k t o w a r d s a d e n s i t y t h e o r e m which i s c o m p a r a t l e t o t h e c e l e b r a t e d one i n r i n o t h e o r y due

t o N . J a c o b s o n . We r e a l l y q e t o n e f o r 2 - p r i m i t i v e n e a r - r i n q s w i t h i d e n t i t y (4.52).

A d d i n q a c h a i n c o n d i t i o n , we a r r i v e a t

a Wedderburn-Artin-like

s t r u c t u r e theorem (4.60).

Before that,

we g e t " b e t t e r a n d b e t t e r " d e n s i t y - l i k e s t r u c t u r e t h e o r e m s f o r 0-,

1- a n d 2 - p r i m i t i v e n e a r - r i n g s .

theorems on v - p r i m i t i v e symmetric v - p r i m i t i v e

I t comes o u t t h a t m a n y

n e a r - r i n q s c a n be d e r i v e d f r o m z e r o -

n e a r - r i n q s w h e r e t h e y a r e much e a s i e r

t o o b t a i n s i n c e t h e s e ones behave more l i k e r i n q s . many p r o o f s c o n c e r n i n g e v e n z e r o - s y m m e t r i c

However,

near-rinqs

differ

t o t a l l y f r o m t h e comparable ones i n r i n g t h e o r y . Anyhow,

t h e " b u i l d i n g stones" mentioned above ( 2 - p r i m i t i v e

n e a r - r i n g s w i t h i d e n t i t y ) a r e shown t o b e d e n s e i n or

Maff(r) ( i f

MGo"k51 ( r ) t M c ( T ) fixed-point-free

No

i s a ring) or i n

( i f No

MGo,{al

i s a non-ring),

automorphism group

where

(r).

AutN 0

HomD(r,r)

(r)

or

Go

i s the

I n particular,

4a General

103

i f Go = ( i d ) , t h e l a t t e r t w o o n e s a r e Mo(r) a n d M ( r ) . F i n a l l y , t h e d e n s i t y p r o p e r t y i s s e e n t o be a k i n d o f a n i n t e r p o l a t i o n p r o p e r t y and a " p u r e l y i n t e r p o l a t i o n - t h e o r e t i c " r e s u l t w i l l be o b t a i n e d . R e c a l l a g a i n ( p . 1 ) t h a t r * = r \ { o ) , a n d so o n .

a 1

G E N E R A L

1 . ) DEFINITIONS A N D E L E M E N T A R Y RESULTS 4.1

CONVENTION I n a l l w h a t f o l l o w s , w w i l l be a n y n u m b e r unless otherwise specified.

E I O , ,Zl ~ 4.2

DEFINITION

( a ) 1.4 i s c a l l e d u - p r i m i t i v e on

Nr:

Nr

is f a i t h f u l

o f type w.

y:

( b ) ri i s v - p r i m i t i v e : 3 N r E N N i s w - p r i m i t i v e on ( c ) I A N i s c a l l e d a u - p r i m i t i v e i d e a l o f N : N/I w - p r i mi t i ve . 4.3

Nr. is

T h e n the following

PROPOSITION L e t ' I be an i d e a l o f N . conditions are equivalent: ( a ) I i s w-primitive. (b)

3 NrENq:

I =

(o:r)

( c ) 3 L s$, N : I = ( L : N ) Proof.

( a ) ->

Nr is

A A

o f type w.

L i s v-modular.

( b ) : I i s w - p r i m i t i v e -> N/I i s u - p r i m i t i v e N / I r -> Nr ( a s i n 3 . 1 4 ( b ) ) i s o f t y p e w a n d

o n some I = (o:r).

( b ) -> ( c ) : L e t r be = Ny p I o l . ( 0 : ~ )= : L . Then 'L L i s m o d u l a r . By 3 . 4 ( e ) , N / L r , so L i s w-modular. Finally, I = ( o : r ) = (o:N/L) = (L:N). nN

( c ) => ( a ) : T a k e N / L = : r . T h e n ( a s above) I = (L:N) = ( o : r ) .

Nr

i s o f type w and

8 4 PRIMITIVE NEAR-RINGS

104

4.4

COROLLARY

The f o l l o w i n g c o n d i t i o n s a r e e q u i v a l e n t :

(a) N i s w-primitive.

4.5

(b)

I01

(c)

3

i s a w-primitive ideal.

L 3% N :

t w-modular A (L:N)

IOI.

=

REMARKS ( a ) Observe t h a t ( c ) i n 4.3 and 4 . 4 q i v e " i n t r i n s i c " characterizations o f primitivity

-

t h a t w i l l be

f o r i t e n a b l e s one t o r e c o g n i z e

extremely helpful,

p r i m i t i v i t y " w i t h i n N". (b) I f

N i s v - p r i m i t i v e on r t h e n IA N

if

i s a w-primitive

(c) 2-primitivity

Nv

( d ) The n e a r - r i n g s

(e)

ideal then

r

Io?

1 4

N.

and

i m p l i e s 1 - p r i m i t i v i t y and t h i s i n t u r n

implies 0-primitivity

n e a r - r i nqs.

N $. f 0 1 ,

( a l w a y s o n t h e same g r o u p ) .

o f 3 .8 a r e e x a m p l e s o f w - p r i m i t i v e

(on Z4).

If N i s v-primitive

on

r

then

M(r)

N 4

(1.48).

( f ) See 95 o f B e t s c h ( 3 ) f o r a d i s c u s s i o n o f t h e s p a c e s o f w-primitive 4.6

PROPOSITION L e t i d e n t i t y e.

( v = 1,2)

ideals

of

NEWo.

N contain either a l e f t or a riqht

Then

(a) Every u - p r i m i t i v e i d e a l I o f N i s modular.

(b) I f e i s a l e f t i d e n t i t y o f N t h e n N i s 1 - p r i m i t i v e iff N i s 2-primitive

(and i n t h i s case e i s a two-

sided identity). Proof.

( a ) I f e i s a l e f t i d e n t i t y i n N t h e n (because

N/I

i s w - p r i m i t i v e o n some

of

N/I

by 3.4(c).

So

N/Ir) e + I

\f nEN: e n

f

i s an i d e n t i t y

n e E n {mod I ) .

If e i s a r i g h t i d e n t i t y , t h e a s s e r t i o n i s t r i v i a l . (b) L e t N be I - p r i m i t i v e on 3.4(c),

Nr

e i s a two-sided

i s unitary.

Nr.

By

i d e n t i t y f o r N.

Flow a p p l y 3 . 7 ( c )

By 3 . 4 ( b ) ,

and 3 . 1 9 ( a ) .

105

4a General

4.7

PROPOSITION L e t FI be s i m p l e a n d N i s v - p r i m i t i v e on r. P r o o f . (o:r) 4 N , so the c o n t r a d i c t i o n

4.8

Nr

(o:r) = to1 Nr = { o l ) .

b e o f t y p e w . Then

(for

(o:r) = N

gives

PROPOSITION ( B e t s c h ( 3 ) ) . L e t t h e r i n g N be w - p r i m i t i v e on r . T h e n N i s a p r i m i t i v e r i n g on t h e N-module r ( ( N . J a c o b s o n ) , p. 4 ) .

Proof, I f

r

= Ny

abelian. If nEN:

then

% r -N N/(o:y)

nl+(o:$),

and

(r,+)

is

n 2 + ( o : y ) ~ N / ( o : y ) then

n(nl+(o:y)+n2+(o:y))

= n ( n , t ( o : y ) ) + n ( n 2 + ( o: y ) )

= nnlt(o:y)+nn2t(o:y)

=

.

v

Hence tf y l , y 2 ~ r nEN: n(y1ty2) nyltny2, and N r i s a ( r i n g - ) module. Each N-submodule o f N r i s a n i d e a l , s o = l o ) o r = r . F i n a l l y , Nr p { o ) by a s s u m p t i o n , s o Nr i s i r r e d u c i b l e a n d N i s p r i m i t i v e on r . 4.9

C O R O L L A R Y (Ramakotaiah ( 1 ) ) . I f N i s commutative and u - p r i m i t i v e then N i s a f i e l d .

LEX~(N):

(L:N) = {OI. L 9 N , s i n c e P r o o f , BY 4 . 4 ( ~ ) , 3 N i s c o m m u t a t i v e . By 3 . 2 5 , (L:N) is the greatest i d e a l i n L, s o L = (01 a n d N c o n t a i n s a r i g h t i d e n t i t y . By 1 . 1 0 7 ( c ) , N i s a r i n g , h e n c e a p r i m i t i v e r i n g by 4 . 8 a n d by ( N . J a c o b s o n ) , p . 7 a f i e l d . In ( 3 ) , R a m a k o t a i a h s h o w s t h a t i f I q N = N o a n d I €g0(t\1) t h e n I is a 0 - p r i m i t i v e ideal. Near-rinas N with a f a i t h f u l , s i m p l e , n o n t r i v i a l N-group a r e c a l l e d ? - p r i m i t i v e a n d a r e s t u d i e d in H a r t n e y ( 4 ) , M e l d r u m ( 7 ) , ( 1 3 ) . S e e a l s o B e i d l e m a n (7),(8),(9). H o l c o m b e blalker ( 1 ) s t u d y 3 - p r i m i t i v e n e a r - r i n q s PI, w h i c h m e a n s t h a t N h a s a f a i t h f u l N-group o f t y p e 3 ( s e e t h e l a s t l i n e s o f p. 80).


106

2 . ) T H E CENTRALIZER

4 . 1 0 DEFINITION E n d N ( r ) = H m N ( r , r ) =: C N ( r ) = : C i s c a l l e d t h e c e n t r a l i z e r o f Nr ( c f . K a a r l i ( 2 1 , R a m a k o t a i a h ( 3 ) ) .

A u t N ( r j =: G o .

A u t N ( r ) =: G N ( r ) =: G;

0

Go:

if

=

~ E C ; ;

l i k e w i s e .:G

otherwise 4 . 1 1 REMARKS ( a ) (C,.)

C Q = {o} ( b ) ~ E (c) If

(ci,o)

. N 4 M C N ( r ) (l-1 5

is f a i t h f u l then

NT

4 . 1 2 NOTATION

a n d (Go,") are q r o u p s , ( " g r o u p s w i t h z e r o " ) a r e rnonoids.

i s a monoid, a n d (Go,,")

(GO,")

If

fn:

nEN,

r

-+

'I

Y

-c

ny

;

FN(r):=

M&9.

ffnlnENl =: F .

4 . 1 3 PROPOSITION ( M l i t z ( 3 ) ) .

(a) If

= ImcM(r)I

(b)

tl

hECN(r):

(c) If Proof.

tl

CN(r)= fom) = : MF(r).

Nr i s m o n o g e n i c t h e n

Q =

r

(a) If YET:

WfEF: mof = h/n = id.

then

CN(r) = { i d } .

hECN(r)

and

fnEFN(r)

then

(hofn)(y) = h(ny) = nh(y) = ( f n o h ) ( y ) ;

so

hEMF(r).

sMF(r). I f y 1 , y 2 & r = Ny a n d y 1 = n l y A y 2 = n 2 y . Then

C o n v e r s e l y , l e t f be

n&N

then

f(nv1) =

3

nl,n2EN:

(fof,)(vl)

=

(fnof)(vl)

= nf(Y1)

and

107

4a General

Hence

feCN(r).

(b)V

heCN(r)

noc:R: h ( n o ) = n h ( o ) = n o .

( c ) f o l l o w s from ( b ) . 1.14 N O T A T I O N

eo:

=

e o ( N r ) := { y c r

el:

=

el(Nr):

Ny = NO =

Rl.

ri.

= C Y E ~ NY =

1 . 1 5 REMARKS ( B e t s c h ( 6 ) ) .

(a)

oEeo,

e l $. 0 ( c ) eon e l (b)

P 0.

so

eo

Nr i s m o n o q e n i c . n $. r .

= 0

r

( d ) Nr i s s t r o n q l y m o n o g e n i c -> ( e ) Nr

eo

i s u n i t a r y ->

and

=> y = l y c R

( f ) G ( B o ) = Bo g r o u p s on

=

A G(B1)

eo

and

(for

= R

w =

= Bow

el,

y e e 0 ->

Ny

R ==>

nocR => Nu = NnocNo = R =I>

so G i n d u c e s p e r m u t a t i o n

el, (if

WE^^)

+ 0)

el

on

el.

The n e x t p r o p o s i t i o n i s a " S c h u r - t y p e lemma". 4 . 1 6 P R O P O S I T I O N ( B e t s c h (S), M l i t z ( 3 ) ) . ( a ) Nr i s s i m p l e (hEC A 2

(b)

n

A

yEB1:

= { o } ->

h(y)EOl)

Nr i s N - s i m p l e

->

C = Colu M o n N ( r )

he6.

->

C = Epi,,,(r,Q)u E p i N ( r , r )

EpiN(r,n) = {el!). (c)

Nr i s N o - s i m p l e

=>

C = G0

and

.

(if

NEW,,

84 PRIMITIVE NEAR-RINGS

108

Proof. ( a ) follows from the f a c t t h a t h s C : Ker h so e i t h e r Ker h = Col ( t h e n h s M o n N ( r ) ) o r Ker h = r ( t h e n h = 0 ) . We may assume t h a t I f hEC A ] yeel: h(y)sel then h 9 6 , so h s M o n N ( r ) . Now h ( r ) = h(Ny) = Nh(y) = r . ( b ) V hsC: Im h = r .

Im h sN

r,

so e i t h e r

Im h = R

SIN

r

r,

+ Iol.

or

( c ) f o l l o w s from ( b ) . i n which We a r e mainly i n t e r e s t e d i n t h e c a s e t h a t C = G o , e v e r y n o n - z e r o N-endomorphism o f r i s an N-automorphism. 4.17 PROPOSITION (Betsch ( 6 ) ) .

( a ) G i s f i x e d - p o i n t - f r e e ( 1 . 4 ( b ) ) on e l . (b)

Nr i s s i m p l e t h e n

If

P r o o f . ( a ) Assume t h a t f o r

'd

6Er

3

= ny = 6 .

( b ) ->:

Then

nEN:

So

6 = ny.

C = Go qEG

Then

and

1

A =

g = id.

Assume t h a t h i s an N-isomorphism r + b c N h s C = G 0 c {6lu A u t N ( r ) , a c o n t r a d i c t i o n .

r.

I f hEC, h 6 then Ker h 9 r . s o Ker h = I o l . ?r T h e r e f o r e h i s a monomorphism and r Irn h . So Im h = r , a n d h s A u t N ( r ) . (0:y)

?

are exactly the o r b i t s

r.

on

Go

%

y%6 -> NO

=

(for

p 6

(o:q(y))

%

% %,%

0

via

No

i n s t e a d o f N stems

f r o m 4 . 1 3 ( c ) : i n t h e f r e q u e n t case t h a t R = o t h e r w i s e be t h e a l l - r e l a t i o n i n a n y c a s e . 4.21 P R O P O S I T I O N (Betsch ( 6 ) ) . % t h e n ?, a n d % c o i n c i d e o n Proof.

If

nly

y%6

If

h : 'I ny

+

+

r

= ( o : 6 ) =>

N = No

nl,n2~N n16

i s well defined.

= n26.

h turns out

n6

t o be an N-automorphism,

Now

would

el.

= n 2 y ==> nl-n2E(o:y)

Therefore

r, ?

Nr i s u n i t a r y a n d

then f o r a l l

(y,6Ee1)

g ( y ) = 6 ==>

gEGo:

=> y a ) .

= (0:s) NO

The r e a s o n f o r d z f i n i n g

3

y%6 ->

so

h ( y ) = h ( 1 y ) = 1 6 = 6,

hEG. hence

y

5

6.

84 PRIMITIVE NEAR-RINGS

110

3 . ) INDEPENDENCE A N D D E N S I T Y

An a p p r o p r i a t e frame f o r o u r

n e x t c o n s i d e r a t i o n s i s g i v e n by

4 . 2 2 D E F I N I T I O N ( M l i t t ( 9 ) ) . L e t M be a n a r b i t r a r y s e t a n d t h e s e t of a l l f i n i t e s u b s e t s o f M . A map f(M) r : f ( M ) + INo i s c a l l e d a r a n k map i f

(a) r(0) = 0 (b)

1

(c)

ti F€f!(M)

FE$(M)

mEM:

ti

r(F u E m l )

m,ncM:

= r(F)+a

with

aEI0,l)

[ r ( F u ( m l ) = r ( F W I n 1 ) = r ( F ) ->

r ( F w { m , n I ) = r(F)].

->

F i s then c a l l e d r-independent i f

r(F) = IF[.

4.23 R E M A R K

With r e s p e c t t o r - i n d e p e n d e n c e , S t e i n i t z ' s t h e o r e m i s f u l f i l l e d ( s e e A . Kertesz, "On independent s e t s of

e l e m e n t s i n a l g e b r a " , Acta S c i . M a t h . 260 - 2 6 9 ) . S e e a l s o K a a r l i ( 2 ) .

(Szesed) 21, 1960,

4.24 EXAMPLES

r ( F ) : = I F [ . Then r i s a r a n k f u n c t i o n a n d every ( f i n i t e ) subset i s r-independent.

( a ) Define

( b ) Take a v e c t o r s p a c e M o v e r a f i e l d K .

Set r(F): = = dim L ( F ) ( l i n e a r h u l l ) . r i s a rank function and r-independence i s j u s t l i n e a r independence.

( c ) Take a n N - g r o u p r a n d d e f i n e f o r e a c h

r(@) a s t h e number o f non % - e q u i v a l e n t g e n e r a t o r s ( i . e . r ( Q ) = l @ n 6 1 / Q l ) . Then r i s a r a n k f u n c t i o n a n d #

= {y l y . . . , y n l

i s r-independent

if

oEf(I')

#eel

tr i S j : v l h j . This independence i s c a l l e d %-independence. The same c a n be d o n e f o r 5.

and

111

4a General

In the theory of rings each primitive r i n g R i s isomorphic t o a " d e n s e " s u b r i n g T'? o f a r i n g HomD(I',I') f o r some i r r e d u c i b l e R-module I' a n d w i t h D = HomR(I',I') ( t h e c e n t r a l i z e r ) making Dr i n t o a v e c t o r s p a c e ( s e e ( N . J a c o b s o n ) , p . 26 3 1 ) . D e n s i t y means h e r e ( i n o u r n o t a t i o n ) t h a t SEN ( y l , ...,y s l lin. indep. i n r 1 61 6 5 ~ r 3 FeR j i E : ( l , sl: r(yi) = 6i.

-

...,

,...,

( I t i s c l e a r t h a t o n l y v a l u e s o f i n d e p e n d e n t e l e m e n t s c a n be arbitrarily prescribed.) We a r e g o i n g t o p r o v e s i m i l a r t h e o r e m s f o r n e a r - r i n g s . B u t b e f o r e d o i n g s o we h a v e t o t a k e a l o o k a t t h e d e n s i t y c o n c e p t ( s e e a l s o Adler ( 1 ) and Ramakotaiah-Rao ( I ) ) . 4 . 2 5 N O T A T I O N L e t M be a s u b s e t o f some a t o p o l o g y i n M as i n B e t s c h ( 7 ) :

We i n t r o d u c e

M(T).

I f mcM a n d y c r , d e f i n e S ( m , y ) : and q : = fS(m,y)[mcM A y E r 1 .

= Im'EM(m'(y)

= m(y)l

4 . 2 6 P R O P O S I T I O N ( B e t s c h (7),(11)). (a)

yis

t h e s u b b a s e o f some t o p o l o g y t o p o l o g y " ) on M.

i s dense i n M

( b ) NEM

V

SEIN

tf m e M

w.r.t.

7

yl, . . . , y

7

(the "finite

s ~ r3

nEN

V

ie(l,

...,$ 1 :

: n ( q ) = m(v+

P r o o f . s t r a i g h t f o r w a r d and hence o m i t t e d . I n a l l t h a t f o l l o w s , " d e n s i t y " means " d e n s i t y w i t h r e s p e c t t o o f 4.26". 4.27 REMARKS ( a ) I f M and N a r e s u b n e a r - r i n g s o f M ( r ) t h e n i t i s easy t o see t h a t No i s d e n s e i n Mo i f f No+Mc(r) i s d e n s e i n Mo+Mc(r). N o t e t h a t N o t M c ( r ) a n d M o + M c ( r ) a r e no n e a r - r i n g s i n g e n e r a l ( s e e 4 . 5 3 ( e ) ) , e x c e p t i n some i m p o r t a n t s p e c i a l c a s e s . ( S e e 4 . 5 4 and 4 . 6 0 . )

112


( b ) If N i s d e n s e i n i n Mo.

M then

9 {id)

( c ) Observe t h a t i f H

r

automorphism group of

tf

b

mEMH(r)

= NnMo(r)

i s dense

i s a fixed-point-free

MH(r) r

then

(since

Mo(l')

h(m(o)) = m(o)).

hEH:

If H = { i d )

No:

then

M(r).

MH(r) =

( d ) We w i l l b e m a i n l y i n t e r e s t e d i n n e a r - r i n g s w h i c h a r e d e n s e i n M (r) a n d 77 (I-):= M (r)+Mc(r) G: G: G: ( 4 . 5 2 and o t h e r s ; . 4 . 2 8 T H E O R E M ( R a m a k o t a i a h (2), B e t s c h ( 7 ) ) .

L e t H be a f i x e d -

r.

p o i n t - f r e e g r o u p o f a u t o m o r p h i s m s o f some g r o u p

(a)

(b)

W

YET*

A

v

ti

3

6Er

mEMH(r):

~ ' E P \ H ~ m: ( y ' ) =

tf S E N b

y1 y

0).

., Y ~ E ~ *

. .

,

Hyi

6 1 y . . . y 6 s ~ r 3 mEMH(r) (c) If

H

6

b

SEN

tf

{id),

W

Mc(r)

(d) I f

3

61y

Proof.

W

nEN

c RH(r),

N

v

V SEN

v

E

iE{l,...,sI:

, Hri

Y~,....Y,E~*

Hyi

...,d S € r 3

nEN

Hyj

tj i c { l , . . . , s I :

I n a n y c a s e we may assume t h a t RH(r) = MH(r)

otherwise (a)

aEHy

-fl

haEH:

Define

mEM(r)

by

a = h,(y)

m(y)

= 6

and

0

mEMH(r);

d e t e r m i n e d by t h e c o n d i t i o n s ( v y ' ~ r \ H y : m(y')

9j

= bi.

HH(r) for

i

+j

= 6i.

{id),

for

(since H i s fixed-

haV)

m(a):= I

clearly

n(yi)

n(yi) H

i

for

M(r).

point-free).

A

=+

= 6i.

m(yi)

Hyj

=/=

N i s dense i n

ylY...,ysEr,

j

M H ( r )

... ,s?:

i E { l ,

i

for

Hyj

i s dense i n

NrMH(r)

61y...y6s~r

= 6 A

(m(y)

= 0).

m(y)

aEHY

.

Then a&HY m i s uniquely = 6 A

113

4a General

( b ) D e f i n e maps

mi€MH(r)

b

= o

mi(y')

y'4Hyi:

m: = m l +

Then

with

...+m S

w i l l do t h e j o b .

By ( b ) a n d 4 . 2 6 ( b ) .

mi(yi)

for

i

9

j,

b

mEMH(r)

the result i s clear.

= m(yi)

nEN: n ( y i )

a>

and t h e r e s u l t f o l l o w s a g a i n f r o m

= m(yj)

4.26( b ) . ( d ) +: By 4.27,

I f one

( s a y yl)

yi

= bi-6

V

fulfills

ncEMc(r) Take

= 61.

...,sl.

for

i ~ { 2 ,

ic{l,.,.,s):

Two o r m o r e

take

= 0,

map w h i c h i s c o n s t a n t no(yi)

n(yi)

=

MH(r).

t o be t h e

n0ENo

with n: = notn

Then

C

= tii.

c a n n o t be z e r o .

yi

(R,,(r))o

i s dense i n

No

If all

the

0,

yi

r e s u l t follows from ( c ) .

Then

r

t r i v i a l l y hold f o r

i s finite).

since

MH(r),

as i n

r. H = {id).

S o we assume t h a t

MH(r)Em0.

(b): Trivial,

0.

Because o f

i s dense i n

4.26.

Proof. Again the r e s u l t s

j

g,,(r).

rev,

Let

n(yi)

for

Hyj

Hyi

MH(r) f

FIH(r).

I14

§4 PRIMITIVE NEAR-RINGS ( b ) ->

( c ) : Assume t h a t H h a s i q f i n i t e l y many o r b i t s

r.

on

3

Then

If

msMH(r) a n d a n e i g h t o u r h o o d U o f m.

Take

3

SEN

y1

= HyJ

Hyi

assume t h a t

,..., y S ~ F :U

then Hyi

S(m,yi)

+ Hyj

for

S 3

fl

i = l

S(m,yi). So we w i l l

= S(m,yj).

i

p

j.

S i n c e H has i n f i n i t e l y many o r b i t s ,

3

y S t l E r \ ( { o l u Hyl u

\

Then

...,s t l l

i t C l ,

ml:

Define

...

=

0

3

Hy,).

...t e s )

m (el+

\y'

eiE:MH(I')

and

jE.Il,

...,s t l l :

= mltestl.

m2:

S

Then

and

ml

If m ( Y s t l ) m1

P0

are

E

fl S(m,yi) i=l

then

ml(Ys+l)

then

m2(yStl) = Y

= 0

5 U.

=I= m(yst,),

= m.I

If m ( y S t 1 ) = so

m2

m2

Anyhow,

0

~

+

+ 0 ~= m

so

(

~

dp m . U c o n t a i n s an e l e m e n t

+m

and

c a n n o t be

discrete. ( c ) ->

(a):

I f H h a s o n l y f i n i t e l y many o r b i t s o n

t h e n each element o f

MH(r)

and o f

flH(r) i s

u n i q u e l y d e t e r m i n e d b y i t s e f f e c t o n f i n i t e l y many s u i t a b l e elements o f So

7is

discrete.

r.

r

~

~

115

4b 0-primitive near-rings

b l 0-PRIMITIVE N E A R - R I N G S

Now we s h a l l p r o v e a " d e n s i t y - l i k e "

s t r u c t u r e theorem f o r

0 - p r i m i t i v e n e a r - r i n g s , Gle s t a r t w i t h z e r o - s y m m e t r i c o n e s . We may a s ~ u m e ( 1 . 4 6 ) t h a t i f N i s 0 - p r i m i t i v e o n r t h e n N Q M ( T ) . G e n e r a l i z a t i o n s c a n b e f o u n d i n M l i t z (4),(3),(12) a n d K a a r l i ( 6 ) .

4 . 3 0 T H E O R E M ( B e t s c h ( 6 ) ) . L e t NewO be 0 - p r i m i t i v e o n r . I f N i s a r i n g t h e n N i s a p r i m i t i v e r i n o on t h e N-module

r and Jacobson's d e n s i t y theorem i s a p p l i c a b l e . I f N i s a n o n - r i n g t h e n we q e t a k i n d o f a d e n s i t y property:

...,y , ~ r , i c { l , ..., s ? : nyi

(D): \ scIN

3

ncN

Proof.

61,...,6s~r

6i.

N i s a r i n g we o n l y h a v e t o a p p l y 4 . 8 .

If

Now l e t l e t

V

%-indep.

yl,

N

s be

I n t h e t e r m i n o l o g y o f (D),

be a n o n - r i n g . > 1

and f o r

...,s - 1 1

tc{l,

let

S(t)

be t h e statement

v Lemma.

t

...,S I : fl

kc{ttl,

(0:Yi)

$ (0:Yk)

,

i = l

Vtc{l,

...,s - 1 1 :

S(t).

P r o o f . By i n d u c t i o n o n t . Since f o r

(o:y)

YEel

V

i d e a l o f N,

i,jE{l,

i s a maximal l e f t

...,s l :

(o:yi)~

E ( o : ~ . ) =>

( o : y . ) = ( o : y . ) => yi%yj -=> i = j . J 1 J Particularly: S(1): k~(2, s l : (o:yl)$

v

...,

9(O:Yk). Now assume

S(t),

s 2 3

and

...,sl.

k~ftt2,

t

Then Since

fl

(O:Yi)$(O:Yk) i=l (o:yk)

and

(o:Yt+l)$(o:Yk)*

i s maximal,

t

[I

(o:yi)+(o:Yk)

i=1

= (o:Yt+l)+(o:Yk)

= N*


116

t

n

Since N i s not a r i n q ,

n (o:yttl)+

(o:vi)

i=l by 3 . 4 ( i ) , $(o:yk) than S f t t l ) .

which i s n o t h i n q e l s e

Now r e t u r n t o t h e p r o o f o f 4.28 and l e t yl, . . . , y , , be a s i n (D). A g a l n we u s e i n d u c t i o n o n 61,.,.,6 S

tc{l,

...,s l .

t = 1

If

then

b

Now assume t h a t

b

iE{1,

..., t l :

3

ylEel

nlaN:

tE{l,,..,s-ll

nlyl

3

= 61.

ntEN

= 6i.

ntyi

t

By t h e lemma,

+ {ol.

Lyttl

3

Therefore

Now we t a k e

: nttlyi

n

L: =

i=l

Since

LYttl

RcL: Lyt+l nttl:

= 6i,

(o:vi)$(o:y

$N

t+l),

hence

(3*4(a)),

= 6ttl-nt6t+l

=

LYttl

r.

.

and g e t \1 i E { l , and t h e p r o o f i s c o m p l e t e . = Ltnt

...,t t l l :

4.31 R E M A R K S

(a)

(D)

i s no " r e a l " d e n s i t y p r o p e r t y s i n c e t h e r e i s no

n e a r - r i n g i n s i g h t i n which f i n i t e topology). ( b ) From

and

(D) i t

N i s dense ( w . r . t t h e

+)

f o l l o w s (Ramakotaiah ( 2 ) ) t h a t ,

yl,...,yscr

if

SEIN ?r

a r e %-independent,

i=l

= W(Yi))). ( c ) The c o n t e n t o f 1Bli

= 1,

(0)

m i g h t be v e r y t h i n :

i f e.g.

(D) i s t r i v i a l . So i t i s n o t t o o s u r p r i s i n g

t h a t t h e c o n v e r s e o f 4.28 does n o t h o l d :

+)

( B e t s c h ) : If o n e c h a n g e s o f k.25 t o Y':={S(m,y)ln&:Mn A y€e,(r)] t h e n o n e g e t s a " r e a l " d e n s i t y t h e o r e m w.r.t. t h e r e s u l t i n g c o a r s e r topology.See also 9.230.


117

L e t N be t h e n o n - r i n q {fEMo(iZ4)lf(2)E{0,21Af(3) 3f(l)I. In z4, O 1 = { 1 , 3 1 , 1 % 3 %( D ) i s f u l f i l l e d , b u t I0,21 QN Z4, so Z4 i s n o t s i m p l e a n d t h e r e f o r e N i s n o t 0 - p r i m i t i v e on Z 4 .

( d ) ( D ) i s equivalent t o t h e following property: (0’):

3

v

yl,. . . , y ,cr,

SEN

iEI1,

nEN

%-indep.

mE:M(r)

.,.,sI: nyi = m ( y i ) .

Now we t u r n t o a r b i t r a r y n e a r - r i n n s . 4.32 T H E O R E M

( a ) Let N be 0 - p r i m i t i v e o n r. Case 1: Nor -/= { 0 1 . Then N o i s 0 - p r i m i t i v e on so 4.30 i s applicable ( f o r r ) , and NC Case 2 :

5

NO

Mc(r).

Nor

(01.

r,

Then

Mc(r)

N =

and

r

i s a

non-zero simple g r o u p i s 0 - p r i m i t i v e on r a n d ( b ) Conversely, i f e i t h e r N o N, E Mc(r) o r i f N = Mc(r) where r (01 i s s i m p l e t h e n N i s 0 - p r i m i t i v e on Proof.

( a ) Anyhow,

If

Nor

I01

N,

e

Mc(r).

r

then NO

and

No

i s 0 - p r i m i t i v e on

(01,

r.

i s o f t y p e 0 by 3 . 1 8 ( a )

r

(3.15(a)).

“by f a i t h ” , s o R = r a n d N = N, = Mc(r) by 1 . 5 0 ( b ) . S i n c e Ncr i s s i m p l e i f f r i s s i m p l e , ( a ) i s shown ( o b s e r v e t h a t Ncr Col!). If

Nor

=

No

= {01

+

( b ) A g a i n b y 3 . 1 8 ( t h i s t i m e by ( b ) ) , i f

N r i s o f t y p e 0 . S i n c e NccMc(r), N o a n d N, ( a n d hence N ) a c t f a i t h f u l l y o n r , s o N i s 0 - p r i m i t i v e o n I’. I f N = Mc(r), r $. Iol and s i m p l e , t h e r e s u l t i s clear.

0 - p r i m i t i v e then

118

4.33

§ 4 PRIMITIVE NEAR-RINGS

R E M A RK

(D) w o u l d

and i n

No

= (0:6)

r,

( i n

if

n o t n e c e s s a r i l y mean t h e same i n would be d e f i n e d by

%

Nr

(o:y)

y%fi:

=

il). Cf. 4 . 1 9 .

4 . 3 4 THEOREM ( R a m a k o t a i a h ( 1 ) ) . E a c h 0 - p r i m i t i v e prime i d e a l

ideal i s a

dp N.

P r o o f . L e t I b e a 0 - p r i m i t i v e i d e a l o f N. L e t Nr b e o f t y p e 0 w i t h g e n e r a t o r yo s u c h t h a t I = ( 0 : r ) (4.3). Assume t h a t

Ji $ (o:r),

J1J2

Jir SO

9 N: J1J2tI

= JiNy

(o:r) = I,

=

c Jiyo

+ {ol.

= Jiyo

Jiyo

0

r.

Now

J1 $ I

A 5

A J2$I.

Jir.

Since

By 3 . 4 ( a ) ,

r,

= Jlr =

JlJ2r

a contradiction.

R E MA R K I n 5 . 4 0 we w i l l s e e t h a t t h e c o n v e r s e o f 4 . 3 4

holds i f 4.36

r.

dN

Jiyo

4.35

J1,J2

i ~ { 1 , 2 1 , Jir

For

so

3

N = No

h a s t h e DCCN.

THEOREM ( R a m a k o t a i a h ( 1 ) ) . E v e r y m a x i m a l m o d u l a r i d e a l

I of

Nsno

Proof.

i s a 0 - p r i m i t i v e one.

L e t I b e a m o d u l a r m a x i m a l i d e a l . By 3 . 2 2 ,

I is

c o n t a i n e d i n a modular maximal l e f t i d e a l L . Since (L:N)

i s the largest ideal of N contained i n L

( b y 3.25), we g e t

by 3.21(a). By 4 . 3 ( c ) ,

I?(L:N)

and

(L:N)

By t h e m a x i m a l i t y o f I,

I i s 0-primitive,

i s modular

I = (L:N).

s i n c e b y 3.29 L i s

0-modul a r . By t h e w a y ,

if N i s 0 - p r i m i t i v e on

n o t necessarily simple (K.

r

a n d Ny = : A t h e n A i s

Kaarli).

F o r t h e r e s t o f t h i s s e c t i o n , we g i v e a d e s c r i p t i o n o f a c l a s s o f 0 - p r i m i t i v e near-rinqs which are n o t 1 - p r i m i t i v e . This d i s c u s s i o n i s d u e t o H o l c o m b e (5), w h e r e t h e p r o o f s c a n b e found,

too.

119


4.37

4.38 DEFINITION I f

:'G

=

EFr, H < A u t ( T ) ,

(r,+)E(a,

t h e t r i p l e (T,B,H)

be the s e t of

r\el

NOTATION I f N r E N q , l e t A : = generators". If A sN r , let (cf. 3.14(a)!).

-

"non-

AutN/(o:A) ( A )

H ( B ) ~ B we , ,-all

c o m p a t i b l e i f a t l e a s t one o f t h e

following conditions i s satisfied:

r.

( a ) B i s no n o r m a l s u b g r o u p o f

3 YEAB 2 (3 h ' c H 3

(b) (c)

(3

A

4.39

BEB

tl

y~r\B

+

hEH : y+B h(y). R E B : y+B = h ' ( y ) ) h

1

Y ' E ~ \ B: h ' ( y ' ) - y ' & B ) .

be 0 - p r i m i t i v e on r , N a n o n - r i n o w i t h NE% i d e n t i t y a n d DCCL, a n d l e t A ( a s i n 4 . 3 7 ) b e an N - s u b g r o u p

THEOREM L e t

of

r

Then

such t h a t

i s not f a i t h f u l , b u t o f type 2.

NA

N i s n o t 1 - p r i m i t i v e on A,

many o r b i t s o n el,

G ( 4 . 1 0 ( b ) ) has f i n i t e l y

i s c o m p a t i b l e and

(T,A,G)

(where A i s a f i n i t e dimensional v e c t o r space o v e r the d i v i s i o n r i n g

GAvta)).

Conversely:

4.40

THEOREM L e t subgroup.

r

Let

b e an a d d i t i v e g r o u p and A be a n o n - z e r o

GA

be a r e g u l a r group o f

automorphisms

o f A w h i c h h a s o n l y f i n i t e l y many o r b i t s o n A. L e t H b e a subgroup o f (a)

( b ) each

Aut

(r,+)

such t h a t

i s compatible.

(r,A,H) hEH

i s r e g u l a r on

AA.

( c ) H h a s o n l y f i n i t e l y many o r b i t s o n (d)

hEH: h/,EG*.

Then

N = IfEMHu{a) ( r ) I f / A E M G A ( A ) l

0-primitive, a n d t h e DCCL.

b u t n o t 1 - p r i m i t i v e on

nA.

i s zerosymmetric,

r,

has an i d e n t i t y


120

r9

I f moreover

and A i s a f i n i t e d i m e n s i o n a l v e c t o r

A

tJ

s p a c e o v e r some d i v i s i o n r i n g D a n d i f then

N = If€M,,"~,)(r)lf/,EEndD(r))

r,

b u t n o t 1 - p r i m i ti ve on

"(

NE

hEH:

h/AED

i s also 0-primitive,

no,?i!z 3, , a n d

moreover

i s a ring.

o :A)

4 . 4 1 R E M A R K See a l s o H o l c o m b e ( 4 ) f o r t h e m o r e q e n e r a l c a s e t h a t A i s o n l y a f i n i t e u n i o n o f N-subgroups o f t y p e 2 w i t h zero intersection. 4.42

t h e n i n t h e n o n - r i n g case o f 4.30

GA = { i d )

REMARK I f

we g e t n e a r - r i n g s o f t h e f o r m N = ( f c M o ( r ) l f ( A ) F A ] ( s e e e . g . N o i n 3 . 8 ) . C f . R a m a k o t a i a h - R a o (1),(3),(4). Conversely,

(r,t)

if

i s a f i n i t e group and A a non-

t r i v i a l subgroup then

N:= { f E M o ( r ) l f ( A ) F A }

i s a finite

n e a r - r i n g w i t h i d e n t i t y , z e r o - s y m m e t r i c a n d 0-, b u t n o t 1 - p r i m i t i v e o n r. A i s j u s t t h e s e t o f n o n - g e n e r a t o r s and i s an N-subgroup s u c h t h a t N / ( o : A ) i s a non-ring if

lAI

> 2.

C\

1 - P R I M I T 1 V E N E A R - R I FIGS

Now l e t N b e 1 - p r i m i t i v e o n

r r

r.

C = Go

Then

(by 4 . 1 8 ) ,

i s n o t N-isomorphic t o a p r o p e r subgroup (4.17(bl),

= eouel

R = Col

(by 4.15(d)),

+

Ak N , L EOI 3 Y E r : We s t i l l a s s u m e t h a t N E

t~ L

4.43

Ly =

or

r

R =

r

(3.2)

and

(by 3.4(a):.

M(r).

THEOREM

r . Then Nor f ( 0 ) A R = r . T h e n N o r , Nc = Mc(r) a n d e l = r .

(a) L e t N be 1 - p r i m i t i v e on C a s e 1:

on

If

No

i s a r i n g then

i s 1-primitive

N i s dense i n

Maff(r)

where r i s a v e c t o r space o v e r t h e d i v i s i o n r i n g D : = HomN (F,r). 0

121

4c 1 -primitive near-rings

If

No i s n o t a r i n g t h e n applicable.

(D)

Case 2 : Nor ( 0 1 A R = ( 0 1 . Then m i t i v e on r a n d 4.30 h o l d s . Case 3: Nor = { o l . Then N = N, a s i m p l e g r o u p dp (01.

=

o f 4.30 i s

N = No

Mc(r)

i s 1-priand

r

is

( b ) C o n v e r s e l y , i f a n e a r - r i n g NEM(r) i s such t h a t N o i s 1 - p r i m i t i v e on r w i t h Nce(IO1, M c ( r ) l o r i f N = Mc(r) (r { o } a n d s i m p l e ) then N i s 1 - p r i m i t i v e on r .

+

+

P r o o f . ( a ) I f Nor (01, N o i s 1 - p r i m i t i v e on r by 3 . 1 8 ( a ) . S i n c e e a c h s t r o n g l y monogenic N-group has e i t h e r R = ( 0 1 o r n = r, t h e r e s t f o l l o w s from 1 . 5 0 , 3.9, 3 . 1 5 ( a ) , 4 . 2 7 ( a ) a n d 4 . 3 2 . ( b ) I f No i s 1 - p r i m i t i v e on r and N, = {O) o r 'L N, Mc(r) t h e n e i t h e r R = ( 0 1 o r R = r ( 1 . 5 0 ) , s o N i s 1 - p r i m i t i v e on r by 3 . 1 8 ( b ) and 3 . 1 5 ( a ) . 'L If N Mc(r), r s i m p l e and (01, then c l e a r l y

-

-

N i s 1 - p r i m i t i v e on

r.

+

4.44 R E M A R K 4.43 i s t h e main r e a s o n f o r d e f i n i n g " s t r o n g l y monogenic N-groups r" as i n 3 . l ( b ) a n d n o t by t h e c o n d i t i o n s "monogenic" and ycr: (Ny = R v Ny r)", f o r 4 . 4 3 would n o t be t r u e i n t h i s c a s e : Take 'l = H8, N o : = { f ~ M ~ ( r ) l f ( =P )f ( 6 ) c 1 0 , 2 , 4 , 6 1 A N,: = {fcMc(r)lf(0)c{0,2,4,611. A f ( 4 ) ~ { 0 , 4 ) ) and

'v

T h e n one can show t h a t N : = N o t N c i s a subnear-ring of M ( r ) enjoying the following properties:

Nr

r

and

a r e f a i t h f u l , s i m p l e a n d monogenic. Moreover,

NO

ycr: (Ny = R = { 0 , 2 , 4 , 6 ) v Ny = r). B u t (01 f R r, and N o i s n o t 1 - p r i m i t i v e on r ( i t i s n o t e v e n t r u e t h a t f o r a l l y c r Noy i s e i t h e r = { o l , = R o r = r, s i n c e N04 = { 0 , 4 1 ) .

122


From 4 . 3 0 a n d 4.45

r

= e0u

el

we g e t w i t h a s t r a i q h t f o r w a r d p r o o f

N€TIO

THEOREM L e t t h e n o n - r i n a

r

be 1 - p r i m i t i v e on

but

without %-equivalent generators.

N

Then

i s dense i n t h e n e a r - r i n q

€370

For 1-primitive near-rinqs

{ f c M o ( r ) \ f ( e o ) = {oil.

w i t h D C C we g e t a w h o l e b u n c h

o f i m p o r t a n t r e s u l t s ( c f . Rarnakotaiah ( 3 ) , B e t s c h ( 1 0 ) ) : 4 . 4 6 THEOREM ( B e t s c h (3)).

Let

e n d o w e d w i t h t h e DCCL.

be 1 - p r i m i t i v e on 7 and

NEWO

Then

( a ) T h e r e a r e o n l y f i n i t e l y many % - e q u i v a l e n c e c l a s s e s i f N i s a non-rinq. (b)

3

S E N: ,,N

s. =

N-isomorphic

1 Li,

r)

l e f t i d e a l s and N-groups o f t y p e 1 ( s o

2.50 i s a p p l i c a b l e ! ) ; s =

f i n i t e l y many p a i r w i s e ( t o

Li

i = l

i f N i s a non-rinq then

Ir/%l-i.

( c ) A l l N - g r o u p s o f t y p e 0 a r e N - i s o m o r p h i c a n d o f t y p e 1. (d) N contains a r i g h t i d e n t i t y (not necessarily two-sided). (e)

N is s i m p l e .

( f ) N i s e i t h e r 2 - p r i m i t i v e o n 'I o r t h e r e i s n o N - g r o u p

o f t y p e 2, Proof.

If N i s a ring,

(b)

-

( f) are e i t h e r well-known

o r t r i v i a l . So we w i l l a s s u m e t h a t

No

i s a non-rinq.

( a ) S u p p o s e t h a t t h e r e a r e i n f i n i t e l y many - - e q u i v a l e n c e assume t h a t

i 2 1,

y o ~ e o . Then

hence

( b ) Now l e t

Y

~

~

representatives of

YOEeOl

yl,

(o:yo)

Y

.~'yS~

= N

p

We may

(o:yi)

for

So b y ( D ) o f 4 . 3 0

y1yy21...~f31.

(o:yo)=(o:y1)=(o:~y1,y2))3'. d i c t i o n t o t h e DCCL.

.

v,,vl.y2,...

classes w i t h representatives

.

which i s a contra-

. b e. a c o m p l e t e s y s t e m o f

t h e %-equivalence classes w i t h

...,y S c e 1 .

S

Then

n

( 0 : ~ ~= ) I 0 1

i= 1

,

but

123

4c I-primitive near-rings

minimal l e f t i d e a l s . Now a p p l y 2 . 5 0 ( p )

tf

j E { l ,

...,sl:

to get

N =

'.1 L

Since

r

by 3 . 1 0 .

j = lj * ,-b

L j -N

L.$(o:Y~), J

( c ) Holds by t h e p r o o f o f ( b ) and 3 . 1 1 ( a ) . ( d ) By ( b ) a n d 3 . 2 7 ( d ) ,

N1 e. sided. (e)

o f 3.8

shows t h a t e i s n o t n e c e s s a r i l y t w o -

3

If I Q N,

minimal, IE(0:L.) J

N contains a riqht identity

. . . ,s l :

jc(1,

L j n I = C O l . =

{O}

But %

(for

L j -N

LjL)I.

Lj

Since

ILjcInL

F), w h e n c e

j

=(O), I = {Ol.

is SO

( f ) By 4 . 7 o r b y ( c ) .

Note t h a t 4.46(a) i s not v a l i d f o r r i n g s : I f 2 s p a c e IR , c o n s i d e r e d a s a n H o m ( r , r ) - m o d u l e , (xELP)

are pairwise inequivalent w.r.t.

n e a r - r i n g which i s p r i m i t i v e on 4.47

=/=

i s the vector all

(1,x)

Hom(r,T)

i s a

a n d h a s t h e DCCL.

N contains a l e f t identity;

COROLLARY N E ' ~ ' ) ~ . D C C N , PI

r

%,

r

I 9 N,

{Ol. T h e n

(a) N !s

1 - p r i m i t i v e c=+

N i s 2 - p r i m i t i v e

N i s simple.

( b ) I i s 1 - p r i m i t i v e c->

I i s 2 - p r i m i t i v e

I i s maximal.

Proof.

( a ) By 3 . 4 ( c )

and 3.7(c),

2-primitivity coincide.

N i s simple then

If

by 3.4(b)

1 - p r i m i t i v i t y and

I n t h i s case,

I = N

N i s simple.

i s a m i n i m a l i d e a l and

a n d Lemma 3 i n t h e p r o o f o f 3 . 5 4

I : = N; t h e n

A =

r)

(with

N has a f a i t h f u l N-group

Nr

o f t y p e 1, s o N i s 1 - p r i m i t i v e . (b)

follows from (a).

K a a r l i ( 2 ) showed t h a t if N = No i s s i m p l e a n d U i s a m a x i m a l N subgroup of

N w i t h NU

{O} t h e n N i s I - p r i m i t i v e .

K a a r l i ( 4 ) and A d l e r ( 1 ) .

Cf.

also

124


d ) 2-PRIMITIVE N E A R - R I N G S

A g a i n we assume t h a t i f

N i s 2 - p r i m i t i v e on r then

NCM(r).

1 . ) 2-PRIMITIVE N E A R - R I N G S The s t r u c t u r e o f 2 - p r i m i t i v e n e a r - r i n g s c a n b e d e s c r i b e d as follows.

4.48 THEOREM ( a ) L e t N be 2 - p r i m i t i v e on Case 1:

Nor

I01 A R

r. Then = r. Then

No

i s 2-primitive

r,

N, = M c ( r ) a n d e l = r . No i s a r i n g t h e n N i s d e n s e i n M a f f ( r ) (as i n 4.43); i f No i s a n o n - r i n g t h e n (0) o f 4 . 3 0 i s a p p l i c a b l e ( f o r No).

on

If

Nor 9 { o }

A

R = {ol.

p r i m i t i v e on

r

and 4 . 3 0

Case 2 :

Case 3 :

Nor

=

(01.

Then

Then

N

= No

i s 2-

i s applicable.

N = Mc(r)

and

r

is a

c y c l i c group o f prime order. Conversely,

(b)

if

Nce{{O),Mc(r))

No


or if

N = Mc(r)

(r

r

o f prime o r d e r ) then N i s 2 - p r i m i t i v e on The p r o o f i s s i m i l a r t o t h e o n e o f 4 . 4 3

4.49

r

a n d if I A

unless order).

I = Mc(r)

r.

and t h e r e f o r e o m i t t e d .

Betsch ( 7 ) ) . I f N i s 2 - p r i m i t i v e { 0 1 , t h e n I i s 2 - p r i m i t i v e o n r, (where r i s n o t a c y c l i c qroup o f prime

T H E O R E M ( c f . F a i n (1) a n d

on

with

a c y c l i c group

N, I

p

125

4d 2-primitive near-rings

( a ) We f i r s t s h o w t h i s t h e o r e m f o r

Proof.

Ir

Ev d e n t l y , Assume t h a t If

IA =

3

i s faithful.

sI r.

A

Iol

then consider

{01.

~ E A : N6

Ir =

But

NEQ.

Therefore

INACIA =

i s f a i t h f u l . Hence

Iol,

N6 =

NA =

(01,

r

A

.b

(01, and N A =

{Ol,

I

so

NA

If

NA.

since

sN f ,

r. Ir

whence

A = {o}.

If

I A

+ {ol

3

then again

6fA:

I6

=+

Io1.

16 sN r , s o 16 = Consequently A = r , f o r I6cA. T h e r e f o r e I i s 2 - p r i m i t i v e o n r. ~ ( 1 6 )= (NI)~FIG,

Since

r.

( b ) Now l e t N b e a r b i t r a r y . We may assume t h a t

N

No,

=/=

I s excluded.

so case 2 o f 4.48

I f N f a l l s i n t o c a s e 1, No i s 2 - p r i m i t i v e o n r . BY 2.18 , I , = ~n ri, 2 N 0 ' I f I , p (0) t h e n I, i s 2 - p r i m i t i v e o n r , hence

I i s 2 - p r i m i t i v e on r by 3.18(b). I f I, = {Ol t h e n I e N c = Mc(r). S i n c e el = r , 10 AN r . 10 = { o l i m p l i e s t h a t f o r a l l y = n o c r and f o r a l l i E I i y = ino = 0 , so I = I o ) .

Hence

10

T a k e any

3

+ Io)

and so

mceMc(T);

is1 : io = yfr:

10 =

m o =:p. C

r. Because o f

10 = ,'I

u.

= i o = p = mco = mcy, h e n c e i = mc % I = Mc(r). If 3 P E P : r Zp, I is 2 - p r i m i t i v e o n r; i f not, I i s not 2-primitive. Now

iy

-

a n d we g e t

If N i s i n c a s e 3, 4.50

REMARK

I i s t r i v i a l l y 2 - p r i m i t i v e on

4 . 4 9 c a n n o t b e t r a n s f e r r e d t o 0- o r 1 - p r i m i t i v i t y ,

n o t even f o r f i n i t e ,

abelian, zerosymmetric near-rings.

I t i s e a s y t o show t h a t i f N i s e . g . O - p r i m i t i v e o n I 9 N t h e n I r i s f a i t h f u l and monogenic. B u t n o t necessarily simple: T a k e r: = z 8 , A : = IO,2,4,61 a n d E: = I 0 , 4 l . N:

r.

= {ffMo(r)lf(A)rA

A f(5)=f(l)

h

f(7)=f(3)),

r

and

I:= (0:A).


126

N i s 0 - p r i m i t i v e on

r,

but

E d1

r.

Moreover,

s t r i c t l y monogenic and I has a r i q h t i d e n t i t y . e v e n b e 0 - p r i m i t i v e o n some o t h e r g r o u p Assuming t h a t ,

take

and ( 0 : 3 ) 1

I"

Ir

is.

I cannot

=:IyA:

Ir

i n

and p u t

L : = ( 0 : ~ ; ) ~Then . L i s a maximal l e f t i d e a l o f I ( 3 . 4 ( f ) ) L (0:1)1 , L ( 0 : 3 ) 1 ( s i n c e ( 0 : 1 ) 1 and (0:3)1 c a n n o t be m a x i m a l ) . T h e r e f o r e ( o : l ) I + L = ( f l : l ) I+ L = I . b u t ( 0 : l ) n (0:3) = E ~ I c - L , s o I w o u l d h a v e t o b e a r i n g

and

by 3.4(i),

a contradiction.

So p o i n t e d o u t , N N =

A s Y.S.

I

(0:l) n(0:3).

S e e m i n g l y t h e r e i s no "smal l e r " c o u n t e r e x a m p l e t h a n t h a t above w i t h 4096 e l e m e n t s . See a l s o 5 . 1 9 ( a ) . By t h e way, case

o n e c a n u s e Z o r n ' s lemma t o show t h a t i n a n y

I 9 N (I9 IO), N w - p r i m i t i v e )

h a s some I - g r o u p s

o f t y p e v. 4 . 5 1 COROLLARY ( F a i n ( 1 ) ) . L e t P b e a 2 - p r i m i t i v e i d e a l o f NET,. L e t I b e a n o t h e r i d e a l o f N c o n t a i n i n g P . Then P i s a 2-primitive

Proof. I

9 P,

A NIP,

ideal of I . and

NIP

'Ip p I01

and

i s 2-primitive.

Since 'Ip i s 2 - p r i m i t i v e . Hence

P i s 2-primitive i n I.

2 . ) 2 - P R I M I T I V E NEAR-RINGS I.IITH IDENTITY

I n t h i s case,

=

Also,

then

if

N = No

(if %

=

% %

NE~,)

or

el

=

r

( i fN $ % ) .

(by 4.21).

Recall that a 1-primitive near-ring cfl0 with a l e f t identity i s a l r e a d y a 2 - p r i m i t i v e one w i t h i d e n t i t y ( 4 . 6 ( b ) ) . We a r e now i n a p o s i t i o n t o g e t a " r e a l " a n d f u n d a m e n t a l d e n s i t y theorem.


N

N f N, (case 1

No a non-ring

No

(case 2 o f

o f 4.48)

No a r i n q

=

127

4.48)

-

Ma f f ( r )

HomD ( r ,r 1

Dr a v e c t o r s p a c e $. fol

l'fc (r)

M C ( r ) = MC(r)

Go f i x e d - p o i n t -

0

0

r

f r e e on

(b) Conversely,

e v e r y n e a r - r i n g w h i c h i s dense i n Maff(r) o r HomD(r,r) (where r i s a non-zero v e c t o r space o v e r some d i v i s i o n r i n g D ) o r d e n s e i n (r) o r

mc 0

(r)

MC

fixed-point-free

(Go

i s 2-primitive

on 7 )

0

on Proof.

r,

where Co = G

w

0

2 - p r i m i t i v e on note that

= MC 0

(r)

4.28(c)

r,

= MGoV{b}

(r)

since

Y I sN r .

then

(r)

My; 0

o(r)

= M

=

Go

(D)

o f 4.30,

and 4 . 2 7 ( a ) .

Mo(r),

then (D)

of 4.30

No,

implies that

N

which i s t r i v i a l l y dense i n ( s i n c e

(r)

MC

= MCal(r)

=

Mo(r).

0 =/=

i s not a rinq,

on

and t h e r e s u l t f o l l o w s f r o m

equal t o )

If N

No

i s therefore I f No i s a

0

Go = { i d }

in

If

i s fixed-point-free

Go

+ {id}

Go

No

and has an i d e n t i t y .

CvcrJs(Y) =

gE:Go:

If

r

and has an i d e n t i t y

cannot occur.

t h e s t a t e m e n t is c l e a r .

ring,

If

r

( a ) I f N i s 2 - p r i m i t i v e on

t h e n case 3 i n 4.48

1

{61.

apply again 4.27(a).

0

i s dense


128

( b ) Assume now t h a t N i s d e n s e i n H o m D ( r , r ) , where i s a n o n - z e r o v e c t o r s p a c e o v e r some s k e w - f i e l d D.

r

N i s a dense s u b r i n g and t h e r e f o r e a p r i m i t i v e

Then

r i n g o n r . F r o m t h i s we d e d u c e : I f N i s dense i n M a f f ( T ) then

i s dense i n

No

s o r h a s no n o n - t r i v i a l N o - s u b g r o u p s , HomD(r,r), a n d N i s 2 - p r i m i t i v e o n r. I f N i s d e n s e i n M O(r) GO

each dense s u b n e a r - r i n g of t h a t i s t r i v i a l l y 2 - p r i m i t i v e o n r. I f G o {id} t h e n M o(r) = FIG ( r ) a n d 0

GO

4.28(c)

Nr c a n n o t c o n t a i n n o n - t r i v i a l

shows t h a t

N-subgroups. Finally i f

in

(Tic 0

N i s dense i n

(r))o

a non-trivial

MC (r)

then

0

As above,

= MC ( I - ) .

r

No

i s dense

cannot c o n t a i n

0

No-subgroup (or one can use 3 . 1 8 ( b ) ) .

4.53 REMARKS (a) I t i s n o t t r u e t h a t each 2 - p r i m i t i v e n e a r - r i n q w i t h identity, N take

MC(r): and

N: =

=/=

r

No

and

a non-ring,

No

i s dense i n

f i n i t e w i t h I i d l f G s A u t ( r ) , G f i x e d - p o i n t free Nr h a s R = r, s o

F i G ( r ) . Then

CN(r) = { i d ) ( 4 . 1 3 ( c ) ) a n d t h e r e f o r e MC(r) = M(r). But N M ( r ) , s o N c a n n o t be d e n s e i n M ( r ) b y

+

4.29.

This i s a l a t e b u t convincing reason f o r

introducing t h i s crazy

N, ( r ) ,

w h e r e one f i r s t

0

s w i t c h e s down t o

No

(by forming

(r)

Co = End

and

NO

t h e n back up b y a d d i n g a l l o f t h e c o n s t a n t s :

MC(r)

would be t o o b i g i n general. ( b ) 4 . 5 2 ( a ) does n e i t h e r h o l d f o r 0 - p r i m i t i v e n e a r - r i n g s with i d e n t i t y nor f o r 2-primitive near-rings without i d e n t i t y ( n o t even f o r

N =

No

and N f i n i t e ) :

129


r:

n4

N : = {fcMo(T)\f(A)!!A) i s 0 - p r i m i t i v e on r w i t h i d e n t i t y , b u t n o t dense i n M C ( r ) = Mo(r) ( 4 . 2 9 ! ) . M: = { f E M o ( r ) l f ( 3 ) = 0 1 i s If

=

and

A:

= (0,2),

0

2 - p r i m i t i v e on

MC (r)

in

r,

w i t h o u t i d e n t i t y and a g a i n n o t dense

Mo(r).

=

0

(c) A l l 2-primitive near-rinas with i d e n t i t y on w h e r e No ( d ) 4.32,

i s a non-ring,

4.43

and 4.48

Z4,

w i l l b e c l a s s i f i e d i n 4.63.

reduce t h e theorv o f p r i m i t i v e

near-rings t o those o f p r i m i t i v e zero-symmetri c nearr i n g s . We w i l l t h e r e f o r e m a i n l y d e a l w i t h t h o s e o n e s

i n t h e sequel. (e) Recall (4.27(a))

that

Kc

general. H e r e i s some e x a m p l e :

= -x)

r A!

=

i s "only a set" i n

(f) 0

G = {id,-idl

i s a fixed-point-free

IR.

= {o,id,-id}.

C:

XER:

f(-x)

(with

-id(x):

=

autsmorphism group on

Vc(IR)

= {fEM(IR)If(O)

= 0 A

-f(x)).

n If m c ( R ) = : N , t a k e nl: = sin+-pN and n2: = n C o n s i d e r n : = nlon2EM(IR ) . = id+pN. n no = s i n o ( i d t r ) - s i n ( % ) i s n o t an odd f u n c t i o n , t h u s n o t b e l o n g i n g t o MC(IR), w h e n c e ngN a n d N i s no n e a r - r i ng.

4.54

COROLLARY I f N i s 2 - p r i m i t i v e o n

r

with

(r)

AutN

= {id)

0

t h e n N i s dense i n e i t h e r one of t h e f o l l o w i n g n e a r - r i n o s ( n o t a t i o n as i n 4.52): H o m D ( r , r ) , Maff(r), Mo(r) o r

M(r) 4.55

(cf.

4.65).

THEOREM ( R a m a k o t a i a h non-ring on

r

classes w.r.t.

(2)).L e t

NET),

be a 2 - p r i m i t i v e

w i t h an i d e n t i t y . Then any two e q u i v a l e n c e

2

(except t h e zero class) are equipotent.

$4 PRIMITIVE NEAR-RINGS

130

T*/%

Proof. L e t E be i n

C o n s i d e r t h e map

and

a f i x e d element o f E.

E

f: G

+

E

9

+

q(E)

(with

Since G i s fixed-point-free

(4.52),

aqain).

f i s injective.

so f i s a b i j e c t i o n .

f i s surjective,

By d e f i n i t i o n ,

G=AutN(r)

3 . ) 2 - P R I M I T I V E Z E R O - S Y M M E T R I C N E A R - R I N G S WITH I D E N T I T Y A N D A M I N I M A L LEFT I D E A L . 4 . 5 6 THEOREM ( B e t s c h ( 7 ) , c f . D e s k i n s ( 2 ) ) . w i t h i d e n t i t y which i s 2 - p r i m i t i v e on

r

L e t N = N , b e a nr. and has a m i n i m a l

l e f t i d e a l L . Then %

(a) L = N r' ( b ) 3 e2 = eeL*:

L = Ne = L e

and

morphic t o Proof.

(a) Since

and

+ {ol,

Lr

3

W i t h y as a b o v e , and

m i ni m a l ) . k-%e

Pie

E

i s antiiso-

(eNe;). YEr:

+ {ol,

Ly

so

Ly =

r

Now we c a n a p p l y 3 . 1 0 .

yeel.

= ey

3

(C,,,(T),a)

Hence

(o:y)nL

= Le = L.

eEL":

e y = y.

=

and

e2 = e

{O} a n d L e

By ( a ) ,

Therefore

I01

(since L i s

Le

{Ol.

e2-eEL n ( o : y ) =

SN L,

Since

Le = L .

By L e C N e s L ,

C N ( r ) = CN(L) = CN(Ne)

( i t can

be e a s i l y v e r i f i e d t h a t N - i s o m o r p h i c N-groups have i s o m o r p h i c centralizer-semigroups). For

neN,

consider

d e f i n e d and

t n : Ne xe

+

+

Ne xene

.

i s well-

tn

E C ~ ( N ~ ) .

C o n s i d e r n e x t t h e map

h:

eNe ene

+ +

CN(Ne). tn

If

e n e = erne

so h i s w e l l - d e f i n e d . C l e a r l y , t h e n t n = t,, h i s an a n t i h o r n o m o r o h i s m . I f h ( e n e ) = h ( e m e )

t n = t,

and

Specializing

then

tj X E N : x e n e = t n ( x e ) = t m ( x e ) = xeme. x = :e

we g e t

e n e = eme

and h i s

shown t o b e i n j e c t i v e . Finally,

b

c€CN(IJe)

3

nEN:

c ( e ) = ne.

ene = e c ( e ) = c ( e 2 ) = c ( e ) = ne aet

c ( x e ) = x c ( e ) = xene

tn(xe),

Therefore

a n d f o r a l l X E N we

so c = tn.


131

and h i s s u r j e c t i v e , hence an a n t i i s o m o r p h i s m . COROLLARY ( B e t s c h ( 7 ) ) .

4.57

N ~ r t ) h a s an i d e n t i t y a n d

If

a minimal l e f t i d e a l L t h e n a l l f a i t h f u l N-groups o f type 2 ( i f those e x i s t ) a r e N-isomorphic

(r,

determines t h e p a i r

I f e i s as i n 4.56(b),

t h e group

CN(r))

u n i q u e l y "up t o i s o m o r p h i s m " .

i s a group w i t h z e r o and L = Ne a s a f i k e d - p o i n t -

(eNe,.) a c t s on

(eNe\{O),*)

( t o L ) and N

f r e e a u t o m o r p h i s m g r o u p (by r i g h t m u l t i p l i c a t i o n ) . Hence e " b r i n g s b a c k " some i n f o r m a t i o n o n

4.58

r

REMARK F o r m o r e i n f o r m a t i o n o n t h e s e t o p i c s ( a p a r t i a l c o n v e r s e o f 4 . 5 6 , t h e u n i q u e n e s s o f (r, CN(r)), s e e B e t s c h ( 6 ) a n d 5 7 a ) , i n p a r t i c u l a r 7.5.

4.)

4.59

COROLLARY L e t 2.50

( f o r "),

(2.50(a)

be a 2 - p r i m i t i v e n e a r - r i n g w i t h

NEW,

5

i s applicable,

hence a l s o

a n d G h a s f i n i t e l y many o r b i t s o n

and 4.21),

i s d i s c r e t e "on

r,

w h i c h i s t h e same a s t o s a y t h a t

MG(r)

Z

(4.29).

7a) f o r the information t h a t i f a f i x e d - p o i n t - f r e e

automorphism group H o f

r

h a s f i n i t e l y many o r b i t s o n

MC(r) h a s t h e DCCL. See a l s o K a a r l i 4.60

etc.)

2 - P R I M I T I V E P!EAR-RINGS WITH I D E N T I T Y A N D MINIMUM CONDITION

DCCL a n d i d e n t i t y . T h e n 4 . 4 6

See

9.227.

out of NSM(T). C f .

r

then

( 2 ) and Oswald ( 1 0 ) .

THEOREM ( B e t s c h ( 7 ) ) . L e t N b e 2 - p r i m i t i v e o n r w i t h D C C f o r t h e l e f t i d e a l s o f No and w i t h a n i d e n t i t y . Then N i s equal t o one o f t h e f o l l o w i n g n e a r - r i n g s ( n o t a t i o n as i n 4.52):

NpNo

N - N o

No a r i n g

Maff(r) H o m D ( r , r )

No a n o n - r i n g

Mc ( r ) 0

MC(r)

dimDr f i n i t e Go h a s f i n i t e l y many

o r b i t s on

r

132


P r o o f . f o l l o w s from 4 . 5 2 and 4.59.

Note t h a t

Rc (I') 0

i s a n e a r - r i n g i n t h i s case ( f o r i t equals N ) . 4 . 6 1 COROLLARY I f N h a s a n i d e n t i t y , i s 2 - p r i m i t i v e o n r a n d i f t h e n o n - r i n g No h a s t h e DCCL a n d A u t N ( r ) = ( i d ) 0

N = M(r) ( i f N No) or o t h e r w i s e N = Mo(r). I n b o t h c a s e s , r ( a n d t h e r e f o r e N, t o o ) i s f i n i t e . So t h e D C C i m p l i e s f i n i t e n e s s ! then e j t h e r

4.62

R E M A R K These r e s u l t s i l l u s t r a t e some r e m a r k s i n t h e

preface: w h i l e the "elements o f r i n q theory" a r e r i n g s

o f l i n e a r mappings on r , t h o s e ones f o r n e a r - r i n q t h e o r y a r e n e a r - r i n g s o f a r b i t r a r y mappings ( p e r h a p s w i t h some r e s t r i c t i o n s ) o n

r.

4 . 6 3 THEOREM ( K a a r l i ( 4 ) ) I f

-

then

I a-S z- N

and i f S / I

i s 2-primitive

I d- N .

Proof. Since I i s a 2 - p r i m i t i v e l e f t i d e a l o f S,

I = (L:S)s

h o l d s f o r some 2 - m o d u l a r l e f t i d e a l L o f S b y 4 . 3 . By 3 . 3 4 ,

L 2NS.

Consequently,

Hence

I

=

(L:SINn S

and ( L : S l N 2 N .

I i s an i d e a l o f N .

See a l s o K a a r l i ( 2 ) a n d R a m a k o t a i a h ( 2 ) .

I n t h e l a t t e r paper i t

i s shown t h a t i f N€Hl i s f i n i t e a n d 2 - p r i m i t i v e o n

r,

if N i s a

lrl-1 i s a p r i m e t h e n e i t h e r N=M(T) o r N=Mo(T) n o n - r i n g and i f or ~ N ~ = ~ i r f~ zr ; i s a b e l i a n , N z N rOr h o l d s i n t h e l a s t c a s e ( t h i s r e s u l t can be deduced f r o m 4.55

and 4.61).

133


5.)

AN APPLICATION TO INTERPOLATION T H E O R Y

4.64

reg

DEFINITION I f

and

NEM(r),

N i s said t o f u l f i l l

the f i n i t e interoolation property I f

w

SE:nU

3

ncN

w

v

Y1#...*YSEr,

Yi

sl:

n(yi)

i c { l #...,

t

Yj

9

for

\I

j

61’..-’6sEr

= 6i.

T h e r e i s an o b t r u s i v e s i m i l a r i t y t o t h e d e n s i t y c o n c e p t s .

In

fact: 4.65

Let

THEOREM

N I M(r)

with

N

and

No

No

not a ring.

Then t h e f o l l o w i n g c o n d i t i o n s a r e e q u i v a l e n t :

r*,

( a ) No i s 2 - f o l d t r a n s i t i v e o n

r

( b ) N i s 2 - p r i m i t i v e on

Go = { i d ) .

with

( c ) N f u l f i l l s the f i n i t e i n t e r p o l a t i o n property.

Proof.

( a ) ->

(b):

Nr i s t r i v i a l l y f a i t h f u l , 2 - f o l d

t r a n s i t i v i t y implies 1-fold i n turn that

Nr

{ol.

If

t r a n s i t i v i t y and t h i s

I o l 9 A sN r ,

take

0

Some

6cA”

No6 =

r

and

Then A =

W

ycr

2

n,ENo:

no6 = y .

So

r.

+

9

If Goy 5: G o 6 but y 6 and ( s a y ) 6 o, take 0 . Then noEN 0 w i t h n o y = o A no6 (0:6) , s o y+6 i n (0:Y) NO NO Nor * h e n c e (4.20(c)), a contradiction, Therefore Goy 9 G o 6 some

+

Go = { i d } .

(b)

(c):

b y 4.54

( c ) ->

(a):

trivial.

K a i s e r ( l ) , Lausch ( 5 ) , Ramakotaiah ( 3 ) .

Cf.

and 4.28(d).

P l i t z (12),(13),

P i l z ( 2 5 ) and

134


4 . 6 6 REMARKS

( a ) So i f a n e a r - r i n g N o f m a p p i n g s o n r i n t e r p o l a t e s a t o and 2 o t h e r p l a c e s then N i n t e r p o l a t e s a l r e a d y on an a r b i t r a r y ( f i n i t e ) n u m b e r o f p l a c e s . Compare t h i s w i t h the corresponding "1 i n e a r " r e s u l t i n r i n g theory ( ( N . J a c o b s o n ) , C o r o l l a r y t o theorem 1 on p . 3 2 ) . This i s a " p u r e l y i n t e r p o l a t i o n - t h e o r e t i c " r e s u l t . ( b ) I t c a n be s h o w n t h a t i f N f u l f i l l s t h e f i n i t e i n t e r p o l a t i o n p r o p e r t y a n d 1i-l > 3 t h e n N o i s a non-

ring. 4 . 6 7 C O R O L L A R Y T a k e r = (IR,+). Then a n y o n e o f t h e f o l l o w i n g n e a r - r i n g s and a l l n e a r - r i n g s c o n t a i n i n q o n e o f t h e m h a v e t h e p r o p e r t i e s t h a t Pi i s 2 - p r i m i t i v e on r w i t h N No, Go = { i d ) and No n o t a r i n g :

+

[XI,N 2 :

t h e n e a r - r i n g o f a l l s t e p f u n c t i o n s on IR the subnear-ring of M(1R) g e n e r a t e d by t h e t r i g o n o metric polynomials.

N1: N3:

= IR

F o r a l l o f them f u l f i l l t h e f i n i t e i n t e r p o l a t i o n p r o p e r t y w h i c h q u a l i f i e s them f o r 4 . 6 5 . The a u t h o r h o p e s t h a t n e a r - r i n g s o f i n t e r p o l a t i n g f u n c t i o n s become i n t e r e s t i n g f o r s p p r o x i m a t i o n t h e o r y ( b e c a u s e t h e s e f u n c t i o n s c a n be i t e r a t e d w . r . t . 0 ) . After a l l t h a t complicated s t u f f the r e a d e r will p o s s i b l y a g r e e w i t h t h e a u t h o r t h a t t h e p r i m i t i v e n e a r - r i n q s have s u c c e s s f u l l y r e v e n g e d t h e i r d i s c r i m i n a t i n g name.

,

135

One f i l l s t h e t r a s h i n t o some b a g s W i t h t h e s e one o n l y c a l c u l a t e s . The r u b b i s h w h i c h you s t i l l c a n s m e l l Is o f t e n c a l l e d t h e " r a d i c a l " . T h i s b e a u t i f u l poem d a t e s way back t o 1 9 7 5 . The a u t h o r i s s t i l l i n h i d i n g .

55

RADICAL THEORY

T h i s p a r a g r a p h e q u a l s on h a r v e s t : t h e s t r a i n s o f p r e v i o u s p a r a g r a p h s a r e h i q h l y r e w a r d e d by t h e f a c t t h a t many r e s u l t s o f t h i s 0 5 a r e easy consequences of previous ones ( c f . e.g. 5 . 4 8 o r 55 c ) , d ) ) . A n e a r - r i n q N m i q h t h a v e no f a i t h f u l N-qroup o f t y p e v . The n e x t q e n e r a l c a s e i s t h a t a l l N - g r o u p s o f t y p e v work t o o e t h e r t o be z e r o . N i s t h e n c a l l e d t o get the intersection fl(o:r) " v - s emi s i mp 1 e 'I . Any how, t h is i n t e r s ec t ion " m e a s u r e s 'I h o w f a r N i s away t o be v - s e m i s i m p l e a n d i s c a l l e d t h e v - r a d i c a l I t contains a l l disoustinq guys, f o r factorino out /dv(N). N ) gives a v-semisimple near-rinq N/

2"(N)'

2v(

F i r s t we g i v e s e v e r a l v-modular l e f t i d e a l s . and Between J O ( N ) object the Ne d i s c u s s , when gv

21(N)

21/2(N),

a l l v , ?,(&I Nil

c h a r a c t e r i z a t i o n s of ?"(N), using We q c t immediately. there i s another radical-like i n t e r s e c t i o n o f a l l 0-modular l e f t i d e a l s . i s " h e r e d i t a r y " a n d prove t h a t f o r

=

N i s v-semisimple

@qv(Ni).

30(N)~31(N)~J2(N)

Also f o r

v

1

$. 7 ,

2v(No)'2v(N).

i f f N i s a s u b d i r e c t product o f v-primitive n e a r - r i n q s . With c h a i n c o n d i t i o n s t h i s s u b d i r e c t p r o d u c t becomes a f i n i t e d i r e c t sum a n d we q e t ( 5 . 3 1 ) i n s p e c i a l c a s e s t h a t N i s v - s e m i s i m p l e i f f N i s a f i n i t e d i r e c t sum o f s i m p l e v - p r i m i t i v e n e a r - r i n o s w i t h D C C L . I n 5 . 3 2 we q e t a " W e d d e r b u r n Artin-1 i k e " s t r u c t u r e t h e o r e m f o r v - s e m i s i m p l e n e a r - r i n q s .

5 5 RADICAL THEORY

136

&(N)

c o n t a i n s a l l n i l N-subqroups,

i d e a l s and

qo(N)

r i n g case,

&(ti)

21,2(N)

all nil left

a l l n i l i d e a l s . However, i n c o n t r a s t t o t h e i s not necessarily n i l i f

N i s f i n i t e (cf.5.48).

F i n a l l y we c o n s i d e r t h e n i l a n d t h e p r i m e r a d i c a l o f a n e a r -

r i ng. a ) JACOBSON-TYPE RADICALS: COMMON THEORY

1.)

DEFINITIONS A N D C H A R A C T E R I Z A T I O N S O F THE RADICALS

As u s u a l , l e t N be a n e a r - r i n g and v ~ { 0 , 1 , 2 1 . Recall our c o n v e n t i o n a b o u t t h e i n t e r s e c t i o n o f an e m p t y c o l l e c t i o n o f s u b s e t s o n p a g e 1. 5.1

DEFINITION

Iv(N):

II

=

Nr o f

(o:r)

i s called the

type v

v - r a d i c a l o f N.

5.2

n

THEOREM * d v ( N ) =

I

I v-Dr.id.of

=

L

n

(L:N)

v-mod. l e f t id.of

N

.

N

Proof. 4.3. The r e l a t i o n s b e t w e e n t h e r a d i c a l s a r e e a s i l y d e s c r i b e d : 5.3

PROPOSITION ( a ) + ' J ~ ( N 5) c d l ( ~ ) c r d 2 ( ~ j ) . (b) I f

NE?$

then

Yl(N)

= q2(N).

( c ) I f N i s a r i n g t h e n 'Jo(N) ( J a c o b s o n - r a d i c a l o f N). Proof.

(a):

= Cal(N)

= )a2(N)

=

q(N)

by 3.7(a).

(b) : b y 3 . 7 1 ~ ) a n d 3 . 1 9 ( a ) . (c):

If

Nr

i s o f t y p e v and N i s a r i n g t h e n one i s a n N - m o d u l e . The r e s t i s

sees as i n 4 . 8 t h a t obvious.

Nr

137

5a Jacobson-type radicals: common theory

5.4

T H E O R E M ( B e t s c h (3)).

+0

qu(N)

P r o o f . By d e f i n i t i o n ,

Nr

u

If

=

n

Nr

i s s t r o n g l y monogenic,

o f type v

(0:r)

so

n

L. L u-mod. l e f t id.in N

*dv(N) =

then

=

(o:r). il

But

(o:y),

YCr

where each

i s

(o:y)

N

o r a v-modular l e f t i d e a l

(3.23). Conversely,

3

Then N/L

?, aN

l e t L b e a v - m o d u l a r l e f t i d e a l of N.

NrENlj

r

3

r

yO€r:

(by 3.4(e))

=

A

NYO

L = (o:y0)

i s o f t y p e u.

(3.23).

Hence t h e

a r e j u s t a l l u - m o d u l a r l e f t i d e a l s ( o r = N)

( 0 : ~ ) ' s

and t h e r e s u l t f o l lows. T h i s r a i s e s t h e q u e s t i o n what happens w i t h t h e i n t e r s e c t i o n o f a l l 0-modular (= maximal modular) l e f t i d e a l s o f 5.5

DEFINITION

21/2(N):

n

=

L

N.

*

L 0-mod. l e f t i d e a l o f rt 5.6

REMARK

%I/2(N)

i s o f t e n denoted by

l i t e r a t u r e (see e.g.

m o t i v a t e d by t h e f a c t t h a t

tdlIE(N)

o n l y " h a l f o f an i d e a l " ( a l e f t ,

"D(N)"

i n the

Our n o t a t i o n i s

Betsch ( 3 ) ) .

i s i n qeneral

but not necessarily

a t w o - s i d e d i d e a l ) and b y i t s l o c a t i o n :

Proof. "Jo(~) =

c

Nr o f

n

n

Nr o f t y p e 0

type 0

These (0:~)'s

n

(o:r)

=

Nr o f

n

type 0

n

ycr

(o:y)

(0:y).

ycel(r)

are (as i n 5.4) e x a c t l y a l l 0-modular

l e f t i d e a l s . Hence

';to(N)

c Pl12(N).

i s a t r i v i a l consequence o f 3 . 7 ( a )

alI2(N) and 5.4.

c al(N)

138

5 5 RADICAL THEORY

The f o l l o w i n g r e s u l t comes f r o m F a i n ( 1 ) . 5.8

PROPOSITION [ L 4, N Hence a l ( N ) and

3

h

22(N)

k L 5'dY(N) A vE{~,~)]=+LE]~(N). a r e semiprime i d e a l s .

k E M :

P r o o f . I f Lk2Ldy(N), b u t L $ ' d V ( N ) , then 3 N r E N 9 : Nr i s o f t y p e v a n d Lr f C o ) . S o 3 y c r : Ly $. l o ) . Hence y & e o , s o Y E e l a n d Ly dN r by 3 . 4 ( a ) . Thus Ly = r a n d Lr = r. T h e r e f o r e r = Lr = = 2~ r = = L k r = { o ) , a c o n t r a d i c t i o n .

...

5.9

REMARK I f L

v = 2

i n 5.8, t h e r e s u l t remains v a l i d i f

s Nr .

5.10 C O R O L L A R Y j l ( N ) c o n t a i n s a l l n i l p o t e n t l e f t i d e a l s and c o n t a i n s m o r e o v e r a1 1 n i l p o t e n t N - s u b g r o u p s .

I2(N)

C f . 5 . 3 7 a n d 5 . 4 5 f o r more r e s u l t s i n t h i s c o n n e c t i o n . 5 . 1 1 E X A M P L E S The f o l l o w i n g e x a m p l e s s h a l l show t h a t no two of a(1,2y g e n e r a l l y c o i n c i d e , n o t even f o r

70,

*dl,

a2

f i n i t e zero-symmetric near-rings.

See Betsch ( 3 ) .

G e n e r a l i z a t i o n s c a n be f o u n d i n Meldrum ( 1 3 ) . Nl i s I-primitive ( a ) N1: = r f E M o ( z 4 ) ! f ( 2 ) = f ( 3 ) = 0 ) . on Z 4 h e n c e Adl(Nl) = {O), b u t n o t 2 - p r i m i t i v e , hence by 4 . 4 6 ( f ) = N1. So =k in general.

"d(N1)

gl(N) q2(N)

( b ) L e t N 2 : = t f ~ ~ l O ( ~ , ) ~ f ( 2 ) ~ { O By , 2 }3 }. 3. we know t h a t N 2 i s 0 - p r i m i t i v e , b u t n o t 1 - p r i m i t i v e on r. S i n c e e a c h map € N 2 i s d e t e r m i n e d by i t s e f f e c t on 1 , 2 , 3 , N 2 i s t h e sum o f t h e l e f t i d e a l s L 1 : = ( 0 : 2 ) n ( 0 : 3 ) , L2: = (O:l)n(O:3) and L3: = ( 0 : 2 ) n ( 0 : 3 ) . Since ( 0 : l ) c\ ( 0 : 2 ) n ( 0 : s ) = { a ] , N 2 = L1;L2;L3. The map

q, * Y

*

I 0

L1

with

fy

N 2 - i s o m o r p h i s m . Hence

fY(x): = 2r

L 1 -N2

q.

Y

x f 1

x = l Similarly,

i s an

139

5a Jacobson-type radicals: common theory fb

B4.

L 3 -N2

and L 2 i s an N2-qroup o f t \ / p e 2 .

lL21 = 2 L2+L3,

Therefore

Ll+L3

and

L1+L2

l e f t ideals. Their intersection f s S i n c e N c o n t a i n s an i d e n t i t y , 5.3(b)).

are 0-modular

31/2(N2)

11(N2)=

q2(N2)

B u t each N2-group o f t y p e 2 i s

3.11(a).

Hence

g2(N2) = (0:L2)

(0). (by

kN L 2

by

$. {Ol =

= (0:2)

= Ca1/2(N2). g1(N2)

Observe t h a t

-

=

g2(N2) =

(0:2)

i s not nilpotent

i n s t r i k i n g contrast t o the s i t u a t i o n in r i n g theory!

Compare 5 . 4 5 !

-

( c ) N3

:=

CfEMo(Z,xZ,)If(A)E

A and

(a,O)-(b,O)

E

A

--',

f ( a , O ) - f ( b , O ) c A 1 w i t h A : = {(0,0),(2,0)1 h a s i d a s i d e n t i t y . A l l ( a , 2 ) w i t h a €2Z4 g e n e r a t e t h e N 3 - Q r o u p r : = Z4xZ2, r h a s o n l y Z4x{Ol and A a s n o n - t r i v i a l N3-

20(N3)

s u b g r o u p s , and they a r e n o t i d e a l s . Hence = {(0,0)1. Now A a n d Z,xEOI/A a r e N 3 - g r o u p s o f t y p e 2 . The a n n i h i l a t o r s o f ( a , O ) ( a r - Z n ) , o f ( 2 , O ) and o f ( 3 , O ) + A are maximal m o d u l a r l e f t i d e a l s o f N 3 , t h e i r i n t e r s e c t i o n 0 3 contains /d,,2(N3). B u t D ' - = C O I . I n 5 . 3 7 ( b ) we w i l l s e e t h a t t h i s implies D C ( N 3 ) , whence = D

f COI 5 . 1 2 EXAMPLE

=

Yo(N3).

If

N = Nc

21 / 2

then

2,12(N3)

Cd,(N) = Cdll2(N)

s e c t i o n o f a l l maximal normal subgroups o f radical" o f

(N,+)),

while

'a2(N)

normal maximal subqroups o f

(Apply 3.21(c), ideal i n

2.)

3.29,

N = Nc

= gl(N)

(N,+)

= inter-

('Baer-

= intersection o f a l l

(N,+).

t h e f a c t t h a t each 0-modular

left

i s 1-modular and 5.2).

RADICALS O F RELATED NEAR-RINGS

20,

a2

To b e a b l e t o t r e a t j1a n d j o i n t l y (at least for a w h i l e ) we i n t r o d u c e t h e f o l l o w i n g d e f i n i t i o n w h i c h comes f r o m u n i v e r s a l a l g e b r a ( s e e (Hoehnke) and M l i t z ( 6 ) ) .

$ 5 RADICAL THEORY

140

5.13 DEFINITION A map % w h i c h a s s i q n s t o e a c h n e a r - r i n q N a n

NN)

ideal

o f N i s c a l l e d a r a d i c a l (map) i f f o r e v e r y

N ,N ' ~ r ) : (a) @N/a(N)) (b) I f

1°)

=

heHom(N,N')

5.14 D E F I N I T I O N

(b) 2 - r a d i c a l :

I A N

h(@(N)) c

a(h(N)).

If% i s some r a d i c a l map t h e n

(a)@-semisimple:

If

then

K

and

5.15 P R O P O S I T I O N

E

g(N) =

c->

N

@(N) denote

I f @,is

NEr)

i s called

(0).

= N. {k+IlkEK)

by

a r a d i c a l map a n d

K+I/I.

N,N'

are

E?)

then (a) If

h: N

radical (b) I f (c) (d)

.

9

N'

and N i s n - r a d i c a l

then

i s $2-

N i s a - r a d i c a l t h e n ! IsN: N / I i s R - r a d i c a l .

b I4N: \ IsN

aL(N/I)3z(N)tI/I.

KEN:

R(N/*) =

K/r

a>

KtIz

8(N)).

(e) I f N i s simple then e i t h e r N is%-radical simple. Proof.

N'

(a):

orR-semi-

b y 5.13(b).

(b): b y ( a ) . (c):

Consider the canonical epimorphism

n:

N

+

N/I

a n d a p p l y 5.13(b).

(d): by (c). ( e ) : t h i s h o l d s because

R(N)A

N.

3"'s

I t w o u l d have been s i l l y t o i n t r o d u c e 5 . 1 3 i f t h e would n o t I n f a c t , B e t s c h ( 3 ) h a s shown t h e f o l l o w i n g

be r a d i c a l s .

141

5a Jacobson-type radicals: c o m m o n theory

5.16 THEOREM

For

, N

vc~0,1,21

-+

2v(N)

Proof. Clearly dv(N) A N. Let n + T v ( N ) b e

,.dv'N/.a

(N) )

i s a r a d i c a l map.

and l e t

be a n N -

r

V

g r o u p o f t y p e u . By 3 . 1 4 ( a ) ,

r

is an

N/

a,(N)-

g r o u p o f t y p e v s i n c e E.dV(N) F ( o : r ) . Hence nr = (n+'d,(N))r = l o ) . S i n c e r was a r b i t r a r y , n&n(O:A), where A r a n g e s o v e r a l l N-groups o f type v. T h u s nECaV(N) and n+av(N) = a v ( N ) , s o 5 . 1 3 ( a ) i s shown. T o s e e 5 . 1 3 ( b ) , l e t h be

and

EHom(N,N')

nc~v("l).

Im h = : N " . L e t I' be a n N " - g r o u p o f t y p e u. 2r Since N" = N/Ker h , r c a n be c o n s i d e r e d a s N-group o f t y p e v ( s e e 3 . 1 4 ( b ) ) . T h e r e f o r e nr = { o } . T h i s i m p l i e s t h a t h ( n ) r = n r = { o l e Again,

r

i s a r b i t r a r y , so

h(n)c2v(h(N)).

5 . 1 7 R E E A R Y , F o r >,,(!I) ( v c { 0 , 1 , 2 } ) , 5 . 1 5 ( e ) c a n be i n p r o v e d i f N i s simple then e i t h e r N i s a v - r a d i c a l or v - p r i m i t i v e (since a l l ( o : r ) 9 N). l h e n e a r - r i n g N 1 o f 5 . 1 1 ( a ) i s an example o f a s i m p l e g 2 - r a d i c a l near-rinq. 5.18 T H E O R E M Let I A N be a d i r e c t summand o f -

Then

N.

Y , , ( N ) ~ I E . ]vv(I) h o l d s f o r a l l v ~ ~ O , 1 / 2 , 1 , 2 1( l e t N = N o f o r v = 1 ) .

S u p p o s e t h a t N = I i J . Then N / J I. Take vcIO,1,2}. Each I - o r o u p o f t y p e v i s a n N - g r o u p o f t h i s t y p e b y 3.14 ( b ) . Let be t h e c l a s s o f t h e s e PI-groups. Now

Proof.

? v ( ~ ) = Nr o f A t y p e v( o : r )

1,'~)0 I

Finally, i f w

n o f type =

1/2

A

W

v (O:r+

=

(o:rlb,

and

v

$pi.

t h e n we g e t w i t h 3 . 2 8 a n d 3 . 3 3

3 1 , 2 ( ~ ) In (fl c( N 1 l n L =

G

I

=

-

A

L E .Yo ( N 1

(r,,I)C;-nin1 L do

=

$ 5 RADICAL THEORY

142

EXAMPLES ( a ) L e t N b e t h e n e a r - r i n g N 2 o f 5 . 1 1 ( b ) .

5.19

(=N1 o f 3 . 8 ) .

Since I i s 1 - p r i m i t i v e on

31(N)= ( 0 : 2 ) =I ( 5 . 1 1 ( b ) ) ,

so

b e f i n i t e and A a n o n - t r i v i a l N:={fEMo(r)If(A)cA} a l s o Ex.

5.32

T1(I)c'&(N)n subgroup.

has ~ l ( ~ l ( N ) ) c J 1 ( N )

lo).

I l v y c

r:

r

Then ( K a a r l i ( 9 ) ) . See

i n Meldrum ( 1 3 ) .

j o (N )

i s 0-primi tive, E

But

I. A l s o l e t

( b ) L e t t h e n o t a t i o n and s i t u a t i o n b e as i n 4 . 5 0 .

I f

LetI:=(O:Z)

Z q ,&=)'I(

f(y)

E

=

{O}.

Since N

I contains the n i l potent ideal

I 0 , 4 1 1 . By 5 . 3 7 ( d ) ,

70(I)# { O l ,

whence

~ o ( I ) ' ~ o ( N )A 1 . 5 . 2 0 T H E O R E M L e t Ni ( i E I ) b e a f a m i l y o f n e a r - r i n g s , ~ ~ ! 0 , 1 , 2 } -____a n d l e t N b e t h e d i r e c t sum o f t h e N i l s w i t h N = N o i f v = 1 . Then P roof. LetvdO,1,21. j e c t i o n s , 5.13

. .1(

I f n.:N+Ni 1

denote the canonical

( b ) g i v e s us f o r a l l

g " ( " q v ( N j ) .

Hence

&(N)E.

pro-

i s 1 the inclusion

mi - J " ( N )

G

i E I

4

0

iE1

?"(Nil.

Conversely,

for

each

l e f t i d e a l s Li

o f Ni

3.33.

l e t

For

i E I ,

i d e a l s i n Ni. Li's,

LEJ"(N)

which are associated w i t h L v i a

ii denote

S i n c e L c o n t a i n s t h e d i r e c t sum o f i t s

I t i s n o t known t o t h e a u t h o r i f 5 . 2 0 = O

1'

the s e t o f these l e f t

we g e t

p r o d u c t s and f o r v = 1 / 2 .

N = N

t h e r e e x i s t v-modular

Anyhow,

i s also true f o r direct

one c a n deduce f r o m 5 . 2 0

that

I a i m p l i e s t h a t f o r a l l v ~ t O , 1 , 2 1 we g e t ?"(N)

=

UEA

1' ?"(Ia). I t i s easy t o see t h a t t h e corresponding r e s u l t UEA does n o t n e c e s s a r i l y h o l d i f t h e Icr a r e " o n l y " l e f t i d e a l s . =

143

5a Jacobson-type radicals: common theory 5. -

( 4 ) ) L e t S be an i n v a r i a n t s u b n e a r - r i n g

2 1 THEOREM - (Kaarli (see 1.31)

o f "No.

Then

120)= In particular, particular,

&(N)n s. i f S i s an i d e a l

this holds

we g e t f o r

e v e r y n e a r - r i n g N=N

72( 1 2 ( N ) ) Proof. -__

Sr

Let

be of

2-modular

o f S/L

2 2 2 ( N ) n S.

Then

is,

as a n S-subgroup,

X ( N ) = N

If{O}

as i n 5.13

radical

z(M)=M,

if

s e m i s i m p l e c l a s s i f I A-N ,

R-semisimple.

5.22

(I),

I( K ( N ) ) =

Kaarli

(a)

(2))

(11,

be zero-symmetric

HolcombeWiegandt

and

( v ~ 1 0 , 1 , 2 } ) i s Kurosh-Amitsur t h e sense t h a t

holds f o r every N t

Amitsur,

Holcombe ( 7 ) , ( 1 5 ) ,

(1).

be as

no.

?,,

is"idempotent"in

( b ) By 5 . 2 1 ,

x(I)=I,

i s said to

~-semisimple,implies

(2),(4),(8),(9), M l i t z ( 7 ) , ( 1 1 ) ,

d e f i n e d on

(Kaarli

N

.

N=N0

The m a i n p a p e r s i n t h i s a r e a w h i c h c o n c e r n

THEOREM L e t a l l n e a r - r i n g s i n 5.13

IqN,

R ( N ) a n d if

X(N)=N f o r every nr.

us h e r e a r e Betsch-Wiegandt Walker

radical

H h a s some

homomorphic image M of

t h e o r y i t i s k n o w n t h a t t h i s h o l d s i f f &(N)=N

implies

hereditary have a -

i s

be

n(I)=I.

with

and h(N)=M i m p l i e s

that I

i s c a l l e d a Kurosh-Amitsur

every non-zero

z(N/I)=N/I

radi-

( 7 ) ) o r even o f c a t e -

(e.g.Mlitz

( 7 ) ) . L e t a l l n e a r - r i n g s u n t i l 5.23

(see e.g.Holcombe

From g e n e r a l and

since

results hold f o r other radicals

universal algebras

zero-symmetric.

ideal

2

)2(N)

I n t h i s a r e a we g e t i n t o c o n t a c t w i t h t h e q e n e r a l

c a l t h e o r y of

if

trivial.

t y p e 2 i s an S-group o f t y p e 2.

of

T h i s makes u s c u r i o u s i f s i m i l a r

gories

&(N).

The c o n v e r s e i n c l u s i o n a l s o h o l d s ,

e v e r y N-group

as well.

:

0

t y p e 2 , f r o n i w h i c h we g e t

Hence Nr i s o f

Again i n

r ks S/L, w h e r e L i s a o f S . B y 3 . 3 4 , L zNS. E v e r y

type 2.

l e f t ideal

N-subgroup

=

i n N.

no.

Tv(

2,,(N))

Ad2

=

iff it TV(N)

/63

i s Kurosh-Amitsur. Also, i s KuroshT 3 ( N ) i s d e f i n e d t o b e t h e intersection where

o f t h e a n n i h i l a t o r s o f a l l N-groups l a s t lines of

p . 80).

r

o f type 3 (see the

35

144

RADICAL THEORY

( c ) By e x a m p l e N 1 ) i n t h e " N e a r - r i n g s o f l o w o r d e r " ( A p p e n -

31

i s n o t Kurosh-Amitsur; in Kaarli ( 9 ) i t i s dix), shown t h a t i s n o t Kurosh-Amitsur e i t h e r . ( d ) If i s K u r o s h - A m i t s u r s u c h t h a t R(N)=N=> N={O} for a l l z e r o - n e a r - r i n g s b u t w i t h K(N)=Nf o r some N = N o = k { O ) t h e n

7o

%? c a n n o t have a h e r e d i t a r y s e m i s i m p l e c l a s s

(BetschBut and have h e r e d i t a r y semisimple c l a s s e s ( K a a r l i ( 4 ) , Holcornbe-Walker ( I ) ) .

T3

72

Wiegandt ( I ) ) .

We c o n c l u d e o u r t r o u b l e s o m e t r i p t o r e l a t i v e s o f N b y a consideration of t h e behaviour o f

2,(N)

on t h e one hand

on t h e o t h e r hand. Our f i r s t r e s u l t i s and '&(No), ],,(Nc) an i m m e d i a t e consequence o f 2.18: 5.23 COROLLARY

b

v ~ C 0 , 1 , 2 ) : >,,(N)

T h i s i s n o t v e r y much, compute

indeed.

I t would be f i n e t o be a b l e t o

and /dv(Nc) (5.12!), a s > " ( N ) = ~ , , ( N ~ ) ~ ~ , , ( N ~s)i ,m i l a r t o 5 . 2 0 . T h i s i s n o t t h e case (see a l s o 9 . 7 7 ) : 5.24

via

= (~,(N)),t(~,(N)),.

>,,(N)

Cdv(N0)

perhaps

If N = IfcM(Z4)(f(0) = f(2) = f(3)l. One c a n show t h a t & ( N ) = N, a 2 ( N o ) = No (5.11(a)!), but o n l y o f t h e maps w h ich are constant c o n s i s t s 12(Nc) EXAMPLE

= 0

= 2.

or

F r o m t h a t o n e s e e s t h a t t h e r e i s no o b v i o u s s i m p l e c o n n e c t i o n between ~v(N),&,(No)

IvC

and JV(Nc).

But av(No)

i s always i n

N) :

5.25 PROPOSITION

N))o;

i n particular,

It suffices t o prove the " i n particular",

?"(No)

See 5 . 6 7

>,(No)%(),(

-

],(No)"),(N) Proof.

V W E C O ,1,21:

(t) for

i s t r i v i a l l y contained i n

(22("1))c

No.

for

145

5a Jacobson-type radicals: common theory

3 . ) SEMISIMPLICITY

T h r o u g h o u t t h i s number, l e t v b e

~{0,1,2),

unles

other wi

indicated. 5.26

DEFIfIITION

N i s v-semisimple:

5.27

EXAMPLE N i s v - p r i m i t i v e ->

5.28 -

COROLLARY

N i s $,,-semisimple.

N i s v-semisimple.

( a ) E a c h d i r e c t sum o r d i r e c t p r o d u c t o f v - s e m i s i m p l e near-rings (b) I f

i s v-semisimple.

b

i E 1 : Ni€?),

tl

icI:

then

8

Ni

i s v - s e m i s i m p l e

ic1 Ni

i s w-semisimple.

P r o o f . 5.20. 5.29 T H E O R E M ( B e t s c h ( 3 ) ) .

N

N i s w - s e m i s i m p l e

i s isomorphic

t o a subdirect product o f v-primitive near-rings. Proof. Consider the s e t o f v - p r i m i t i v e 1 . 5 8 and 4 . 2 ( c ) .

5.2,

N have t h e

5.30 T H E O R E M L e t c->

i d e a l s and a p p l y

D C C I (DCCL). Then N i s v - s e m i s i m p l e

N i s isomorphic t o a subdirect product o f f i n i t e l y

many u - p r i m i t i v e n e a r - r i n g s w i t h D C C I (DCCL). P r o o f . =I>: of

The f a m i l y

N has

n

(Io)aEA o f all v-primitive

I a = {Ol.

ideals

We c l a i m t h a t i t s u f f i c e s

UEA

t o t a k e f i n i t e l y many section

I f not,

1"'s

t a k e some

t o get a zero i n t e r I (aEA). Since aO

n aEA

=

CO),

t h e r e i s some

al€A:

I

fi aO

Con t in u n g i n t h i s w a y we g e t a c h a i n

I =I O1 a.

.

I 31 n I a . a. "1

=...

8 5 RADICAL THEORY

146

w h i c h d o e s n o t t e r m i n a t e a n d we a r r i v e a t a c o n t r a d i c t i o n . Hence N i s i s o m o r p h i c t o a s u b d i r e c t p r o d u c t o f f i n i t e l y many v - p r i m i t i v e n e a r - r i n g s . The r e s t f o l l o w s by 2 . 3 5 . f o l l o w s from 5 . 2 9 .

:

by 5 . 3 0 , 4 . 4 6 ( e ) a n d ( d ) a n d 2 . 5 2 ( b ) .

:

&/2(N)

If

t h e 0-modular

= tUI,

the intersection o f

to}.

l e f t ideals =

As i n t h e p r o o f

o f 5 . 3 0 i t s u f f i c e s t o t a k e f i n i t e l y many o f them, s a y K1, Kk, m i n i m a l f o r h a v i n g i n t e r s e c t i o n = {O}. k Apply 2 . 5 0 ( 9 ) t o see t h a t N = l'Li, where i= 1

...,

Li= j$i

( a ) : By 2 . 5 0 ,

( b ) => Li

by 5.32.

(b):

k If

N h a s t h e DCCL.

N-simple l e f t i d e a l s ,

then

N =

Ki:

n

2 - m o d u l a r l e f t i d e a l s and ( a s i n 2.50(9)) hence

22(N)

=

( b ) ->

(c):

If

= COI,

Ki

i = l

101.

N

k =

l'Li i = l

then by (a)

as a b o v e ,

...,k I

there i s a subset S o f

and 5 . 3 4 ( a )

l*Li, i = l

{l,

with

t y p e 2.

r,

If

A sN

$:=

{E AN T I A n E = { o ) ) . But

$Atr.

A n(Liyotr)

F.

maximality o f ( c ) =>

(d)

{ol

=

Hence

{O)

L gL 14:

f(M1)+L

+

M1

f +

i n

LiyO'$

Liyon(AtT)

= Io).

a n d ( c ) i s shown.

NN

i s monogenic (by e ) . be exact.

M2

= N A f(M1)nL

the "projection"

iES:

which c o n t r a d i c t s the

A t ' t T = I'

i s t r i v i a l since

( d ) ==> ( e ) : L e t

3

A+F $; r, 3

If

i s o f t y p e 2, h e n c e

Liyo

Therefore

'E: ( Z o r n ! )

t a k e some m a x i m a l e l e m e n t

= (01.

By ( d )

Then

f(M1) defined by i s an N-homomorphism. p(n) = p(f(ml)te):= f(ml) p/ =:p. Then f'lop: M 2 -+ M1 i s an N-homomorphism

p: N

+

M2

with

= idM ,

f-lopof

Hence

1 ( e ) =>

(f):

and l e t

{Ol

MI

+

Let

M1

1: 1 4

( f ) => map h: M2

(e):

id +

-+

f(M1).

N-homomorphism

M2

splits.

b e as i n t h e s t a t e m e n t ,

+

M1).

Then Then

f M1 + M 2 b e e x a c t . The i d e n t i t y c a n b e e x t e n d e d t o a n N-homomorphism

Let

f(M1)

-+

does t h e r e q u i r e d j o b .

T

+

Ml

b e t h e i n j e c t i o n map.

M2

s p l i t s ( s a y b y g: M 2

M2

5 : = hog: M2

M1,M2,r,h +

{O)

{Ol

+

Then c l e a r l y M2

+

MI.

f-loh

i s a splitting

9 5 RADICAL THEORY

158

( e ) ->

( 9 ) : F i r s t we show t h a t N h a s t h e D C C N . be a c n a i n o f N - s u b g r o u p s o f N . N = f40d~1=42=,..

Let

tl

L

i s m : 10)

+

M~

iM

( L ~t h e

~

-

~

i n j e c t i o n maps)

spl i t s .

Mi be c o r r e s p o n d i n g s p l i t t i n g Nhomomorphisms. Then hl: = g 1 i s a s p l i t t i n g N MI + N, h2: = g20g1 one f o r homomorphism f o r C O I ( 0 ) * M e + N, e c e t e a. I f Li: = K e r h t t h e n Li A, N a n d ( a s e a s i l y s e e n ) Furthermore, N = MitLi with Mi" L = Io).

Let

gi:

Mi-l

+

+

... .

L1"L2" But

i s comp e t e l y r e d u c i b l e ( 2 . 4 8 ( e ) ) ,

NN

eEN),

generable (since (2.50(e))

which causes

f i n i t e l y many s t e p s .

finitely

s o endowed w i t h t h e ACCL

Ll=L2=

...

to stop after

Hence t h e same a p p l i e s t o

.. .

Mo=M1=.

Now we show t h a t N h a s no n o n - z e r o n i l p o t e n t Ns u b g r o u p s . L e t M be such one. As b e f o r e , L n M = {Ol. L 4, N : N = L t M ,

3

Let

be = I l o t m o . ( & , E L ,

e

moe!1). A s i n 3.43,

mo

t u r n s o u t t o be a r i g h t i d e n t i t y f o r M, hence M c a n n o t be n i l p o t e n t and t h e p r o o f i s a c c o m p l i s h e d .

(9) -> ( b ) : N-subgroup.

N = Noeot(O:eo)

By 1.13,

Mon(O:eo) If

Mo = N

Hence Either

( B l a c k e t t ( 2 ) ) . L e t Mo b e a m i n i m a l 3 eo = eocN: 2 Neo = Moeo = Mo.

By 3 . 5 2 ,

= Mot(o:eo)

with

= (01.

t h e r e i s n o t h i n g t o p r o v e . So l e t

N=(o:eo) (o:eo)

Mo

9 N.

$. COI. i s a minimal N-subaroup o r i t c o n t a i n s

( b y a p p l y i n g t h e above c o n s i d e r a t i o n s t o (o:eo) i n s t e a d o f N) a n o t h e r s m a l l e r N - s u b g r o u p o f t h e form

(o:e,)nL

w h e r e L i s some l e f t i d e a l o f

N.

The D C C N a s s u r e s t h a t a f t e r f i n i t e l y many s t e p s we a r r i v e a t a m i n i m a l N - g r o u p intersection o f

(o:eo)

M1

which i s the

w i t h a l e f t i d e a l o f N,

h e n c e a n o r m a l s u b g r o u p . T a k e some i d e m p o t e n t

elcN

5b Jacobson-type radicals: special theory

with

Nel

groups).

= Mlel

Hence

M1.

159

(@:eo)=M,+(o:el)

Repeating t h i s procedure w i t h

(as

(o:el)

...

( i f necesszry) y i e l d s N = Mo: ;Mk w h e r e Mi a r e m i n i m a l (hence N - s i m p l e ) N-subgroups o f N.

Now by 5 , 4 0

21,2(N)

so N i s t h e d i r e c t s u m

IOI,

=

S

I*Li i = l

o f l e f t i d e a l s of type 0 (5.39).

o f above a r e N-groups o f t y p e 2,

The

Mi's

hence N-isomorphic

t o some L . ' s . J T h e r e f o r e N i s t h e f i n i t e d i r e c t sum o f l e f t i d e a l s w h i c h a r e N-groups of 5.50

t y p e 2.

REMARKS

(1) h a v e shown o n e c a n a d d t o t h e f o l l o w i n g c o n d i t i o n i f NE?:l

(a) As Choudhari-Tewari 5.49

( i )E a c h N - s u b g r o u p o f N i s m o n o g e n i c , p r o j e c t i v e ( d e f i n i t i o n a g a i n as u s u a l ) and g e n e r a t e d b y a n idempo t e n t

.

See t h e r e f o r t h e p r o o f (b) If N = N o

( c ) ->

i s 2-semisimple

(i) =>

from 4.46(d),

( 3 ) and S c o t t ( 1 ) .

.

Cf.

Chao ( 1 ) .

t h e n t h e DCCL o r DCCPI i m p l y

a l l o t h e r c h a i n c o n d i t i o n s o f ACCL, This follows

(e)

5.49(d)

ACCN,

DCCL,

and ( 9 ) . C f .

DCCN. Oswald

I t i s n o t known t o t h e a u t h o r i f a

2 - s e m i s i m p l e n e a r - r i n g w i t h ACCL o r A C C N h a s a l l o t h e r c h a i n c o n d i t i o n s as w e l l . ( c ) A s Mason ( 3 ) p o i n t e d o u t , i n j e c t i v e N=No-group.

( 3 ) and ( 4 1 ,

there e x i s t s no n o n - t r i v i a l

See m o r e o n t h a t i n h i s p a p e r s

i n Prehn ( I ) ,

Maxson ( 8 ) and Oswald ( 1 0 ) .

Banaschewski-Nelson I n particular,

(I),

see 9.264.

160

$ 5 RADICAL THEORY c ) THE N I L RADICAL

5 . 5 1 D E F I N I T I O N T h e sum o f a l l n i l i d e a l s o f N i s c a l l e d t h e

n i l r a d i c a l o f N and d e n o t e d by

V(N)

R a m a k o t a i a h (1) and P o l i n ( 2 ) ) .

Cf.

in

'J1(N)

(by

Gojan ( 1 ) .

5 . 5 2 THEOREM (a)

V(N)

i s t h e g r e a t e s t n i l i d e a l o f N. i s t h e s m a l l e s t i d e a l I o f N such t h a t

(b)?l(N)

N/I

has no n o n - z e r o n i l i d e a l s . Proof.

by 2 . 1 0 1 ( b ) .

(a):

n: N

Let

(b):

+

N/n(N)

N

=:

be the canonical

projection. If

If

T

4

IT, T

I : = n-l(f) 1 N. kEIN: n ( i ) = v ( i ) k = U ( z e r o i n

nil,

look a t

k

then 3 k i EKer u = "I(N). But i s n i l , so % S I N : (ikIE = 0 and i i s n i l p o t e n t . Hence I i s iE1

n(N)

hence

3

nil, therefore

It'?I(N)

Now ascurne t h a t By 2 . 1 0 3 ,

I+fl(N)/,

a n d we a r r i v e a t 5.53

COROLLARIES

N/I

a n d we g e t

T

=

{Ul.

i s w i t h o u t non-zero n i l i d e a l s . i s nil in

N/I,

so

I+?1(N)EI

T(N)SI.

(Rarnakotaiah ( 3 1 , Meldrurn ( 7 ) )

( a ) W i s a r a d i c a l map ( i n t h e s e n s e o f 5 . 1 3 ) . ( b ) V(N) 5 TI",) 5 ' d o ( N o ) t Cdo(N). ( c ) N i s n - s e m i s i r n p l e i f f N has no non-zero n i l i d e a l s .

(d) Each c o n s t a n t n e a r - r i n g i s 9 - s e m i s i m p l e . Proof.

(a):

by 2.100 and 5.52(b).

( b ) : b y 2.99, ( c ) : by

5.52(a),

55Z(b).

(d): by (b).

m),

5.37(d)

and 5.25.

161

5d The prime radical

It i s clear that f o r rings

coincides with the usual n i l

%(N)

radical o f rings.

? i sl s u b h e r e d i t a r y

5 . 5 4 THEOREM ( c f .

Maxson ( 1 ) ) . I f

then %(I) c

o n d i r e c t summands:

I 4 N

i s a d i r e c t summand

V(N) A I .

The p r o o f f o l l o w s f r o m 2.12. 5.55

REMARK?l(N)

i s a l s o i d e n t i c a l w i t h t h e "upper n i l r a d i c a l "

U = s(0)

o f Van d e r W a l t ( 1 ) . See t h i s p a p e r f o r a

V(N)

characterization o f

v i a "s-systems".

See a l s o

Beidleman ( 9 ) .

d ) T H E P R I M E RADICAL

5.56 DEFINITION The i n t e r s e c t i o n o f a l l p r i m e i d e a l s o f N i s

P(N)

c a l l e d t h e prime r a d i c a l o f N and denoted b y (other notations: >-*(N), Again,

L,(N),

m(0)).

Cf.

Gojan

(1).

t h i s i s j u s t t h e usual prime r a d i c a l i n the case of

rings. 5 . 5 7 PROPOSITION Proof.

pis

a r a d i c a l map.

( a ) P ( N ) 4 N.

(b) I f P:

h:N 3

showing t h a t

P

and

P 4 ??

i s prime then

i s p r i m e i n N b y 2.64

= h-l($)

(c) If

TT

h(P(N))

and 2.17(a),

cV(T).

i s a prime i d e a l o f

'IT: = N/aO(Nl

i ~ - l ( F )= : P i s p r i m e i n N. i f P 4 N i s prime then

i n (b),

Conversely, i n TI.

n(P)

then,

as

i s prime

H e n c e i f Ti i s i n e a c h p r i m e i d e a l o f TT t h e n e a c h xch-'({T)) and

X

i s i n each prime i d e a l o f

i s zero.

Therefore

?o(f4/aD(N))

i s zero.

N, s o

XERN)

1 62

6 5 RADICAL THEORY

The c o n n e c t i o n t o 2 . 9 3

i s q i v e n by and t h i s i s a semiprime i d e a l .

5.58 R E M A R K p ( N ) = / 8 ( { 0 } )

i s a n i l i d e a l and c o n t a i n s the sun

5.59 PROPOSITIONPP(N)

o f a l l nilpotent ideals. P r o o f : b y 2.105. F r o m t h i s we c a n l o c a t e P ( N ) : ~ ( N ) ~ T ? ( N ) ~ ~ o ( f i )I ! 2+ ()N, ) % j l ( N ) F ) 2 ( N )

5 . 6 0 COROLLARY

(and a l l i n c l u s i o n s can be s t r i c t ) . The f i r s t t w o i n c l u s i o n s c a n e v e n b e s t r i c t i n t h e c a s e o f r i n a s . 5 . 6 1 THEOREM

Let

(a)T(N)

...

-

Proof.

WN)

=

(b)Ea2(N)

N

E =

~h a v e t h e D C C N . T h e n

"d").

is nilpotent (cf. = &(N).

(a) f o l l o w s f r o m 4.34

(b):

"'I:

5.48)

P(N)='h(N)

=

and 5 . 4 0 ( c ) .

i s trivial.

f o l l o w s from ( a ) and 5.48(d).

5 . 6 2 THEOREM ( M a x s o n ( 1 ) ) . I f

'p(1) 5

IA

P(N)n

N

i s a d i r e c t summand t h e n

I.

T h i s r e s u l t f o l l o w s f r o m 2.63. 2.69 y i e l d s 5.63

EXAMPLE E a c h p r i m e ( e . g . semisimple.

More g e n e r a l l y :

each c o n s t a n t ) n e a r - r i n g i s

-

163

5e Concluding remarks

5.64 PROPOSITION N i s

iP-sernisimple i f f N i s i s o m o r p h i c t o a s u b d i r e c t p r o d u c t o f prime n e a r - r i n n s .

T h i s i s a d i r e c t c o n s e q u e n c e of 1 . 5 8 a n d 2 . 6 7 . 5 . 6 5 PROPOSITION Each p - s e m i s i m p l e n e a r - r i n g h a s no n o n - z e r o

n i l potent ideals. T h i s f o l l o w s f r o m 2 . 1 0 4 o r from 5 . 5 9 .

S e e more on t h a t i n S c o t t ( I ) , H o l c o n b e ( 2 ) a n d R a m a k o t a i a h - R a o ( 5 ) .

e ) C O N C L U D I N G REMARKS 5 . 6 6 SUMMARY Gle s u m m a r i z e some p r o p e r t i e s o f o u r r a d i c a l s (we i n c l u d e $1,2 a l t h o u g h i t i s n o t a r a d i c a l map).

Radical

‘8

R(N)q u a s i r e q u l a r ( N ) ? a l l quasi regul ar

’a2

’dl

21/2

I - I - 0

( N ) ?a 1 1 q u s s i r e g u 1 a r

i

%(N)

nil (N)”all n i l N-subgroups a(N)?all nil l e f t ideals (N)?all nil ideals (N)”all nilpotent ideals

-

-

(t)

-

(t)

(+)

t

t

+

t

+

t

( N ) i s the greatest

(N) i s t h e g r e a t e s t

1 - 1 - 1 -

8 5 RADICAL THEORY

164

II + II

me a n s

" y e s 'I

-

me a n s

'I

II

I1

no

"(+)"

stands f o r

NECI)~)"

I'

"yes,

if

( o t h e r w i s e un-

known t o t h e a u t h o r )

If

NE?,

has a minimal No-subgroup t h e n a l l r a d i c a l s a r e

NcWo

h a s t h e DCCN t h e n

=I= N* If

r2(N)

i s n i l iff a l l radicals

are equal.

5.67 SOME MORE REMARKS ( a ) S e e B e i d l e m a n (1),(3),(8),(11) between

rd2(N)

about t h e connection

and " ( s t r i c t l y )

small" l e f t ideals.

S i m i l a r c o n s i d e r a t i o n s can be f o u n d i n R i e d l ( l ) , C h a o and M l i t z ( l ) , ( 2 ) .

(1)

They a d o p t a l a t t i c e - t h e o r e t i c

p o i n t o f view ( t h e i n t e r s e c t i o n o f a l l maximal i d e a l s

(...)

= sum o f a l l " s m a l l " i d e a l s ( . . . ) ) . C f .

Oswald ( 5 ) .

( b ) R a m a k o t a i a h (1) s h o w e d t h a t e a c h b i r e g u l a r n e a r - r i n q

(3.49)

i s 0-semisimple.

( c ) The a u t h o r s u g g e s t s n o t t o u s e t h e n o t a t i o n s for

"I a n d

p,

respectively,

Jacobson-type

because these a r e not

radicals.

( d ) Ramakotaiah (1),(3) a l s o d e f i n e s a r a d i c a l "J-3(H)" c o n t a i n e d i n P(N), a s t h e i n t e r s e c t i o n o f a l l i d e a l s

I such t h a t

N/I

h a s no n i l p o t e n t i d e a l s .

f r o m T h e o r e m 8 o f Van d e r W a l t ( 1 ) t h a t

P(N)

a l l semiprime i d e a l s ( c f .

Maxson ( 1 ) ) and o f

( e ) See F r e i d m a n ( l ) , B h a n d a r i - S a x e n a

f o r a "Levi tzky-type"

2-3 = z-*.

( 1 ) f o r other charac(such as t h e i n t e r s e c t i o n o f

See a l s o t h e p a p e r b y C h o u d h a r i terizations o f

It follows

n(N).

( 2 ) and P l o t k i n ( 2 )

radical.

( f ) Ramakotaiah ( 4 ) a l s o d e f i n e d a r a d i c a l corresponding

t o t h e Brown-Mc-Coy

radical (?-radical)

t h e o r y as t h e i n t e r s e c t i o n modular i d e a l s .

Y(N)

in ring

o f a l l maximal

See a l s o C h o u d h a r i - T e w a r i

(3).

165


If ZEN is called "G-regular" i f the ideal generated {n-nz(nENl equals N then g(N) turns o u t t o be the intersection of all ideals I of N , such that N/I has no G-requl a r has no G-regular ideals. N/9(N) ideals and is a subdirect product of simple near-rinos with a right identity. by

(9) Laxton ( 3 ) defined o n e more "radical-like" ideal S ( N ) of N a s the intersection of all "s-primitive ideals". H e showed that lll2(N) F S ( N ) 5 a l ( N ) if NcV0

and gives a n example o f a dg. near-ring with 21/2(N) = S(N) = 21(N). See also Beidleman (7),(8),(9), Hartney ( 2 ) , ( 4 ) and Meldrum ( 5 ) , ( 1 3 ) . (h) Another radical w a s defined by Deskins (1) (see also Williams ( 1 ) ) . If N = N o has the DCCN then semisimplicity w.r.t. this radical is equivalent t o 2-semisimplicity, and this in turn to semisimplicity in the s e n s e o f Blackett ( l ) , (2) (see 5 . 4 9 ) . (i) Beidleman considered in (2) the "radical subqroup" RS(N) as t h e intersection of all maximal N-subqroups in near-rings By 5.38 w e know that in this case R,(N) 5 Beidleman proved e.g. that RS(N) = 'J2(N) i s quasiregular (in his sense - s e e 3.37(c)). C f . 5.48(d).

€no.

12(N)

g2(~).

Q(N) = n L , where L ranges o v e r all maximal left ideals, w a s also considered (by various authors). If N has a right identity then (3.29) This and m o r e radicals c a n Q(N) = a l I 2 ( N ) be found in Choudhari 1).

(j) T h e "quasi-radical"

(k) Gorton (1) called an N group Nr t o be of c l a s s A ( A a non-zero cardinal number) if A E r . lAllA WfcM(r) 3 ncN V yEr: f(y) = ny. N i s called A-complete if N has a faithful N-group of class A . A radical C,(N) is defined as the intersection of all ( o : r ) , w h e r e r is a n N-group o f class a .

55

166

RADICAL THEORY ?*

He showed t h a t i f N i s A - c o m p l e t e on r t h e n N c a N r , and t h a t N i s 1-complete i f f N c is f a i t h f u l (a base 1.91). If N d 0 then E C1(N). of equality D e f i n i n g CX-modular l e f t i d e a l s a s t h o s e modular l e f t i s a n N-group o f c l a s s X i d e a l s L such t h a t N / L ( c f . 3.28) y i e l d s a r e s u l t s i m i l a r t o 5.2. A l s o , he g a v e s e v e r a l e x a m p l e s .

g2(N)

-

( 1 ) Maxson ( 1 ) p r o v e d t h a t t h e r e i s n o t such a f i n e c o n n e c t i o n between i n j e c t i v i t y o f N-groups ( d e f i n e d a s u s u a l ) and s e m i s i m p l i c i t y a s i n t h e r i n g - c a s e . He showed t h a t i f e a c h N-group i s i n j e c t i v e t h e n b u t g a v e an e x a m p l e t h a t t h e c o n v e r s e = COI, does not hold.

I2(N)

( m ) Ferrero developed a r a d i c a l theory f o r " p - s i n g u l a r n e a r - r i n g s " i n (18).

'a2)

(n) A radical (correspondinq t o f o r N - g r o u p s was c o n s i d e r e d by B e i d l e m a n i n ( 1 ) , ( 3 ) a n d ( 4 ) a n d by Choudhari i n ( 1 ) . (0)

V a n d e r W a l t ( 1 ) c a l l e d an i d e a l I o f N a n i l r a d i c a l i f I i s n i l , b u t N/I h a s no n i l i d e a l s a n y m o r e . He p r o v e d t h a t t h e sum o f a l l n i l r a d i c a l s o f N e q u a l s Q(M), w h i c h i s t h e g r e a t e s t n i l r a d i c a l o f N , w h i l e the i n t e r s e c t i o n of a l l n i l r a d i c a l s ( t h e smallest n i l r a d i c a l ) c o i n c i d e s w i t h P ( N ) . T h e r e f o r e he c a l l e d "I(?')t h e u p p e r ( l o w e r ) n i l r a d i c a l o f N.

( p ) M l i t z ( 2 ) , ( 3 ) and P o l i n ( 2 ) g e n e r a l i z e d t h i s r a d i c a l t h e o r y t o what t h e y c a l l e d "m-R-near-rinqs".

( 9 ) See a l s o o t h e r p a p e r s of M l i t z f o r a r a d i c a l t h e o r y i n u n i v e r s a l a l q e b r a s . However, t h e s e r a d i c a l c o n c e p t s turn o u t t o be " t o o l e s s n e a r - r i n q - s p e c i f i c " . ( r ) Another a t t e m p t t o g e t a r a d i c a l t h e o r y f o r r e r o s y m m e t r i c n e a r - r i n q s was made by S c o t t i n ( 4 ) . He u s e d a method s i m i l a r t o t h a t o f ( D i v i n s k y ) f o r r i n g s . As a n e x a m p l e he s t u d i e s t h e B a e r - l o w e r - r a d i c a l , w h i c h t u r n s o u t t o be = P ( N ) = > - 3 ( N ) for near-rinss w i t h D C C on N - s u b g r o u p s . C f . Holcombe ( 3 ) , ( 8 ) and K a a r l i (4) ,(7).


167

( s ) I t i s e a s y t o see t h a t i f N i s a z e r o - s y m m e t r i c n e a r -

r i n g w i t h i d e n t i t y a n d t h e D C C N a n d i f e i s some i d e m for all p o t e n t i n N t h e n J v ( N ) e = Ne n

Z,(N)

V&{O,

1,Zl.

I f Ne i s a m i n i m a l n o n - n i l p o t e n t N - s u b g r o u p o f N a n d if

z2(N)

i s n i l p o t e n t then

2,(N)e

i s the greatest

p r o p e r N - s u b g r o u p o f Ne, s o Ne/ ZZ(N)e

i s an N - g r o u p

o f t y p e 2 ( i t i s h a r d e r t o see t h a t a l l N-groups o f t y p e 2 a r i s e i n t h i s way).

,

(1)

See L a u s c h - F j o b a u e r

where t h e s e r e s u l t s a r e f o r -

m u l a t e d and p r o v e d f o r d g n r . ' ~ - b u t t h e y a r e v a l i d i n t h e general case. (t) Let,

denote the s e t o f a l l

f o r t h e moment,&

maximal" ( c f .

3.29)

ideals o f

F,oFlc

( i. e .

"strictly

those i d e a l s

of

N w h i c h a r e a t t h e same t i m e m a x i m a l N o - s u b No q r o u p s o f PIc). R o u t i n e a r n u m e n t s c r i v e t h e f o l l o w i n r r

g2(tI))c =

i n f o r m a t i o n on

If

L E

LnX C =

J , ( N )r i~ c: tic o r L nPIC€ 4

Ms?, N, c o n t a i n s L n N c 0 ?+I1 = N a n d Y = " ) .

(for if whence Conversely, Hence

( then

2 2(I.l)

M

if

E&

' d 2 ( N ) nP i c

See a l s o 5.12,

5.23,

n

then

N0tM

E

L+Ms

then

NO

N,

d2(N).

M.

M €4 5.24

and 9.77.

( u ) I n M e l d r u m ( 1 3 ) i t i s shown t h a t

&!(I)

=

mN)nI

if

I i s a d i r e c t summand o f N a n d t h a t

8?(

8 ieI

Ni)

=

8

R(Ni).

I n here,

?I?=$ o r @ = %.

ieI

( v ) In Angerer-Pilz

(1)

i t i s shown t h a t t h e r e e x i s t s a

n e a r - r i n g N o f o r d e r 32 w i t h .k,(N)c~l,2(N)c;\ll(N)C%,(N),

and 3 2 i s t h e s m a l l e s t

o r d e r such t h a t these f o u r r a d i c a l s a r e d i f f e r e n t . (See a l s o Meldrum ( 1 3 ) ) .

Also,

the following results

from Angerer ( 1 ) concerning r a d i c a l s o f " s m a l l " nearr i n g s are mentioned:

85

168

( a ) If ( N , + )

i s simple then e i t h e r

31(N)=

{OI,

3

N o r 1 / 2 ( N ) = C O I , " i j i ( ~ )= N . I N 1 i s t h e p r o d u c t of t w o primes o r i f

&(N)

( B ) If

RADICAL THEORY

=

(N,+)

i s c y c l i c o r n o n - a b e l i a n of o r d e r 8 t h e n

'k0(N)

=

Z,(N).

(Y) I f I N 1 i s t h e p r o d u c t of t h r e e d i f f e r e n t primes

2,(N) form a c h a i n t h e n e i t h e r Io(N)c y,/z(N)c Il(N) Zz(N) N o r Po(N)= T,/z(N)= % l i ( N ) & ( N ) o r 'Jo(N) zIjz(N) a,(H)c 22(N)o r a l l r a d i c a

then = %l/z(N). ( 6 ) I f t h e normal s u b g r o u p s o f ( T i , + )

=

=

=

=

=

S

coincide.

( w ) See t h e " N e a r - r i n g s of l o w o r d e r " i n t h e Appendix o r t h e r a d i c a l s o f n e a r - r i n g s o n m o s t g r o u p s of o r d e r 5 8 .

PART 111 SPECIAL CLASSES OF NEAR-RINGS § 6 D I STR IBUT IVE LY

G E N E RATED N EAR-R 1 NGS

8 7 TRANSFORMATION NEAR-RINGS

58 NEAR-FIELDS AND PLANAR NEAR-RINGS

59 MORE CLASSES OF NEAR-RINGS To keep this monograph within a reasonable size we will only cite, but not give proofs of some statements which lie a l i t t l e bit away from the main stream of discussion (but might be equally important)

170

56

DISTRIBUTIVELY GENERATED NEAR-RINGS

I n t h i s p a r a g r a p h we d i s c u s s t h e s e t y p e s o f n e a r - r i n q s w h i c h a r e s t i l l more " r i n g - l i k e " t h a n z e r o - s y m m e t r i c n e a r - r i n g s . I n f a c t , e v e r y d g n r . i s €3,. I f N i s a d g n r . t h e n t h e i d e a l s o f N T a r e e x a c t l y t h e normal N - s u b g r o u p s , b u t t h i s n i c e f e a t u r e d o e s n o t seem t o h e l p a l o t . F o r i n s t a n c e , a l l n e a r r i n g r a d i c a l s c a n s t i l l be d i f f e r e n t ( e v e n f o r f i n i t e d n n r . ' ~ ) . Abelian d g n r , ' s a r e r i n q s . We a l s o d i s c u s s t h e o p e n p r o b l e m o f e m b e d d i n g a z e r o - s y m m e t r i c near-ring i n t o a dgnr. I n t h e c a s e o f n e a r - r i n g homomorphisms t h o s e o n e s d e s e r v e p a r t i c u l a r i n t e r e s t which c a r r y t h e d i s t r i b u t i v e g e n e r a t o r s i n t o t h e ones o f t h e image. These "(N,D)-(N',D')-homomorphisms" a r e c h a r a c t e r i z e d . A l t h o u g h t h e d q n r . ' s f o r m no v a r i e t y , i t i s p o s s i b l e t o speak about " f r e e n e a r - r i n q s d i s t r i b u t i v e l y g e n e r a t e d by a g i v e n s e m i g r o u p " . N - q r o u p s r a r e s t u d i e d w h i c h have t h e p r o p e r t y t h a t t h e d i s t r i b u t i v e q e n e r a t o r s o f 4 I act " d i s t r i b u t i v e " ( = a s endomorphisms) o v e r r . F i n a l l y we s t u d y t h e s t r u c t u r e o f d q n r . ' ~ : 2 - p r i m i t i v e f i n i t e for a d g . n o n - r i n g s w i t h i d e n t i t y a r e j u s t t h e Mo(r)'s f i n i t e , n o n - a b e l i a n i n v a r i a n t l y s i m p l e group r . In t h e f i n i t e c a s e , Mo(r) = E(r) i f f r i s o f t h i s k i n d .

.

171

6a Elementary

a )

E L E M E N T A R Y

(dq.,b e t t e r :

N i s d i s t r i b u t i v e l y generated generable)

distributively

i f t h e r e i s a subsemigroup D o f

(Nd,.)

qenerating

(N,+). 6.1

NOTATION I f D g e n e r a t e s N we d e n o t e t h i s b y

6.2

EXAMPLES ( s e e ilolcomhe ( 3 ) f o r g e n e r a l i z a t i o n s ) (a)

If

(r,+).c'$l, Cuiei,

where

E(r)

i s a subnear-rinq o f

(H. Let

u i ~ { - l , + l l and

(End r,o)

Neumann ( l ) , ( 2 ) ;

(rn,+)

{ e l,...,enl

M(r),

ei.cEnd

r.

distributively

a n d c a l l e d t h e " e n d o m o r p h i sm

I"' ( s e e 1 . 1 5 ) .

near-ring on (b)

E(r) b y t h e s e t o f a l l f i n i t e

define

sums

generated by

(N,D).

Frohlich ( l ) ,

be a reduced f r e e

=:

E

(i.e.

(2)).

qroup w i t h generators

e a c h map

be e x t e n d e d t o an endomorphism on

E

+

Pn

can uniquely

rn; r n i s

then

t h e f r e e g r o u p i n some v a r i e t y o f g r o u p s ) . Define f o r the set

rnd

rn

two b i n a r y o p e r a t i o n s 8 , -

by

(61 8 6 2 ) ( e i ) :

(61

*

62)(ei):

@l(ei)+62(ei)

= ~ $ ~ ( $ ~ ( e ~( a) n )d e x t e n d f r o m E


172

I n F r o h l i c h ' s p a p e r s , + i s r e f e r r e d t o a s the " a d d i t i o n o f t h e f i r s t t y p e " a n d 13 a s t h e " a d d i t i o n of t h e second type". ( c ) S i m i l a r t o ( a ) , the near-rinqs A(r) a n d I ( r ) , d e f i n e d a s t h e s u b n e a r - r i n g s o f M(r) g e n e r a t e d by the automorphisms ( i n n e r automorphisms) o f ( I - , + ) , are d g n r . ' ~ . 6.3

REMARKS A(r)

(a) E(T),

and

I ( r ) w i l l be s t u d i e d i n 9 7 c ) .

( b ) The E n d r n l s w e r e i n t r o d u c e d a n d s t u d i e d by H . Neuinann i n ( 1 ) and ( 2 ) . Her r e s u l t s on t h e s e t y p e s of near-rings include: E n d r n c o n t a i n s no i d e n t i t y , b u t a l l 0 f i x i n g some e i and s e n d i n g the o t h e r e ' s i n t o z e r o a r e j d i s t r i b u t i v e a n d c a n be v i e w e d a s " r z l a t i v e u n i t s " . T h e r e i s a 1 - 1 - c o r r e s p o n d e n c e rl, b e t w e e n t h e s e t f n of a l l f u l l y i n v a r i a n t subgroups o f r n and the s e t In o f a l l i d e a l s o f E n d r n by $:

Jn A

+

+

zn

bicfl,...,nl:

$(ei)cA1

A l l homomorphic i m a g e s o f End r n a r e a l s o some E n d rm's. Each E n d T n i s t h e homomorphic i m a g e o f End a n , where i s t h e f r e e g r o u p on n generators. S i m i l a r r e s u l t s hold f o r t h e near-rings of t h e kind 8 End r n . which a r e a l s o dg. ( s e e 6 . 9 ( d ) ) . n E IN ( c ) See F i t t i n g ( 1 ) f o r t h e problem, which automorphisms

o f a ( n o n - a b e l i a n ) group have the p r o p e r t y t h a t t h e i r s u m i s a n a u t o r n o r p h i s m a g a i n . C f . a l s o Heerema (1) and Robinson ( 1 ) f o r s i m i l a r q u e s t i o n s . ( d ) See P l o t k i n ( 2 ) f o r t h e c o n n e c t i o n between t h e r e p r e s e n t a t i o n s o f a group r and t h o s e o f E(!'). ( e ) See a l l papers of Dasic f o r g e n e r a l i z a t i o n s o f the conc e p t o f a d g n r . . C f . a l s o Meldrum ( 1 3 ) .

173

6a Elementary

Now we

s t u d y some e l e m e n t a r y p r o p e r t i e s o f d g n r . ' ~ . N o t e ,

i f N i s dg. b y D t h e n e a c h n = Caidi

6.4

with

ncN

= +1,

ai

that

i s a f i n i t e ( o r d e r e d ) sum

di€D.

PROPOSITION L e t N b e d o . b y D. (a) (b) (c)

tl n c N \1

dcD: d ( - n ) = ( - d ) n = - ( d n ) .

NEq).

\

n,n'EN

dsD:

= (-d)n't(-djn

(d) I f

n = laidi i

d(ntn')

= -dn'

and

-

2

dn+dn' A ( - d ) ( n + n ' )

=

dn.

n' = laid;

then

j

nn' = ~ a i ( ~ a ! d . d ! ) . 1

J

3 1 3

The p r o o f i s a c c o m p l i s h e d b y e a s y c o m p u t a t i o n s a n d t h e r e fore omitted.

6.5 PROPOSITION (Seth-Tewari ( I ) , Meldrum (13)). Let N be dg. by D a n d r an N-group with d ( y + y ' ) = dy+dy' for a l l deD, y,y't;D. If A S r then the N-ideal h generated by A is given by all finite sums of the form C ( y i + a i d i d - y i ) with yi", c r i ~ { l , - l l , dieD and 6eA. Proof.

The s e t o f a l l f i n i t e sums o f t h e f o r m X a i d i 6 i i s a

(r,+). h i s then closure o f A i n (r,+). To s e e t h a t A gN r , c o n s i d e r subgroup A of

n as

n = Caidi

a n d 6 as

j u s t t h e usual n o r m a l n(T+y)-ny,

decompose

= x(y.+u.d!G.-y.) 1 1 1 1 1

and

p r o c e e d a s u s u a l . (The n e x t r e s u l t shows t h a t i t s u f f i c e s t o show t h a t NEGK.)

See Meldrum (13) that 6.5 i s not v a l i d without the d(y+y')=dy+dy'assumption. Near-rings generated by a n inverse semigroup of distributive elements are treated i n M a h m o o d - M e l d r u m - O ' C a r o l l ( 1 ) a n d Meldrum (13). Examples of d.g. near-rings of low order c a n he found i n the appendix.

174

$ 6 DISTRIBUTIVELY GENERATED NEAR-RINGS

b)

6.6

SOME

AXIOIIATICS

P R O P O S I T I O N L e t N b e d q . by D a n d

r

( a ) I f A i s a normal s u b q r o u p o f

(I",+)

be a n N-qroup. then

r < = > ~ s

ASI

N N (This i s one s t e p t o w a r d s t h e s i t u a t i o n i n rinos,

Nr

since the ideals of qroups. )

a r e j u s t t h e normal N-sub-

2

( b ) H i s a b e l i a n N i s d i s t r i b u t i v e . + ) ( c ) N i s a b e l i a n c-> N i s a r i n o . ( d ) I f N E ~t h e n N i s d i s t r i b u t i v e N i s a b e l i a n N i s a r i n g . k

P roof. -

(a) If

1aidicN,

n = k

n(b+y)-ny = +...to

1aidi(6+y)-

i=l d ( b + Y )-okdkYk k

Since

dk(Gty)-dky

...-

and

A5

= -dky-dkb+dkyEA,

r

then

~

E

=

aldl(6+y)+

N

k

1aidiy

i=l 1d 1y .

= dk6tdky-dky = dk6Eh

(-dk)(G+Y)-(-dk)Y

6.4)

ycr

i=l

and (USinr:

= (-dk)Y+(-dk)b-(-dk)Y

=

we s e e t h a t i n a n y c a s e

k d ( 6 +y ) - 0 k d k y ~ A . P r o c e e d i n ? i n t h i s way y i e l d s A r. The c o n v e r s e f o l l o w s from 1.34(b) and 6.4(b). ( b ) =>: I f N 2 i s a b e l i a n t h e n f o r n , n ' , n " E N , n = X u i d i w e o e t n ( n ' + n " ) = C q 1. d 1. ( n ' t n " ) = = ~ a . ( d . n l + d . n " ) = ~ u ~ d ~ n ' t C a . d . n n" n ' t n n " . 1

:

N*

c o n t a i n s an e l e m e n t w h i c h i s n o d i v i s o r

o f tero.

( b ) b nEN*:(NEnl

and

nEN

i s invertible)

n i s no

zero divisor.

Proof. ( a ) "->"

a d i v i s o r o f zero. 3 k e I N : Nxk = NXk + l 3 e E N : x . x k = e-xktl. Hence

So a s s u m e t h a t 2 Now N x 4 x ?...

i s clear.

x-ex = 0

--

,.. . So

.

x

(9

0)

Therefore

This implies that ( x - e x ) x k = 0.

a n d we g e t

e x = x.

i s not

179

6d Finiteness conditions

Also, ( x e - x ) x = 0 a n d t h u s xe = x . S o \1 meN: (me-m)x = 0 w h e n c e me = m . Now t a k e some a r b i t r a r y n E N . Decompose x a s x = Coidi. Then x(en-n) = Cuidi(en-n) = = E u i ( d i n - d i n ) = 0 , i m p l y i n g t h a t en = n . (b) Let

i s c l e a r again. n ;P 0 be no z e r o d i v i s o r . T h e n N E ~ , by ( a ) . A s i n ( a ) , 3 kclN : Nnk = N n k + ’ = This i m p l i e s t h a t So 3 meN: n k Ink = ( 1 - m n ) n k = 0 , so 1 = m n . Also, (nm-1)n = 0 , s o n m = 1 and n i s i n v e r t i b l e . “->I’

... .

m a n k t 1 .

6.15 REMARK ( L i q h ( 1 3 ) ) . I f N i s a f i n i t e simple d g . near-ring then (N,+) i s a p e r f e c t ?roup ( i . e . N coincides with i t s commutator s u b g r o u p ) . See a l s o F e i g e l s t o c k ( 2 ) . There a r e s e v e r a l c o n n e c t i o n s between c h a i n c o n d i t i o n s , s o l v a b i l i t y of (N,+) a n d “weak d i s t r i b u t i v i t y ” ( s e e F r o h l i c h ( l ) , D e f . 4 . 3 . 1 ) . We s t a t e w i t h o u t p r o o f t h e f o l l o w i n g c o l l e c t i o n o f r e s u l t s ( s e e a l s o Beidleman ( 1 1 ) ) . 6.16 THEOREM L e t N be a d g n r . ( a ) (Frohlich ( 1 ) ) . I f (N,+) i s solvable then N i s weakly d i s t r i b u t i v e . If N2 = N, the converse also h o l d s . See a l s o Mason ( 1 ) . (b)

(Beidleman ( 4 ) ) . I f N i s f i n i t e and i f NrtzNq t h e n (l‘,+) i s s o l v a b l e i f f N r i s s o l v a b l e ( i . e . r has a normal s e q u e n c e ( 2 . 3 7 ) w i t h a b e l i a n q u o t i e n t s ) .

i s s o l v a b l e a n d N has ( c ) ( B e i d l e m a n ( 4 ) ) . I f (N,+) t h e D C C N t h e n e v e r y maximal l e f t i d e a l i s m o d u l a r a n d c o n t a i n s t h e commutator subgroup o f ( N , t ) . ’;J2(N) i s nilpotent and is a r i n a . Also, N has a 2(N 1 c e r t a i n kind o f ACC.

N/l

(d) (Ligh ( 3 ) ) . I f ( N , + ) i s s o l v a b l e such t h a t n o t a l l elements a r e d i v i s o r s of z e r o . Then t h e DCCL implies the ACCL.

D ISTR IBUTIVE LY GENE RATED NEAR-R INGS

86

180

e ) "FREE" DISTRIBUTIVELY G E N E R A T E D N E A R - R I N G S

Since the dgnr.'s

do n o t f o r m a v a r i e t y ,

f o r t h e existence o f " f r e e dgnr.

I s " .

t h e r e i s no g u a r a n t e e

Moreover,

t h i s concept

does n o t seem t o b e a p p r o p r i a t e f o r t h i s c l a s s o f n e a r - r i n g s . We a r e now g o i n g t o d e f i n e a s i m i l a r c o n c e p t . F i r s t o f a l l we n e e d a " r e f i n e d " v e r s i o n o f homomorphisms b e t w e e n d g n r . I s .

6.17 DEFINITION L e t

6.18

h: N

+

No

h(D)

5

D'.

be d q n r . ' ~ . A h o m o m o r p h i s m

(N',D')

(N,D),

(N,D)-(N',D')-homomorphism

i s c a l l e d an

N

EXAMPLE Each d g n r . - h o m o m o r p h i s m

N'

+

i s an

i f

(N,Nd)-

-(N',N,!,)-homomorphism. 6.19 PROPOSITION ( F r o h l i c h ( 2 ) ) .

and l e t

h:

(N,+)

-c

Let

(N',+)

a s e m i g r o u p homomorphism

be a g r o u p homomorphism and (D,.)

+

(D',.).

(N,D)-(N',D')-homomorphism.

Then h i s an

P r o o f . I t o n l y r e m a i n s t o show t h a t = h ( n ) h ( n' ). Let

n = Caidi,

n ' = Ca'.d!.

1

J

J

1

ioi(lujh(di)h(d;))

n,n'cN: Then,

J J

h(nnl) = h(lai(la!d.d!))

=

(N',D') b e d g .

(N,D).

1

J

J

= ( l o i h ( d i ) ) . ( l o j h ( d ! ) )J

J

1

=

using 6.4(d),

= loi(la;h(did!))

J

h(nn')

= =

J

= h(n)h(n').

We a r e now g o i n g t o d e f i n e s o m e t h i n g l i k e a " f r e e n e a r - r i n g dg. b y a g i v e n s e m i g r o u p

(D,.)".

We u s e a s l i g h t m o d i f i c a t i o n o f a m e t h o d d u e t o F r o h l i c h ( 4 ) and M e l d r u m ( 2 ) . C f . a l s o Zeamer ( 1 ) . 6.20 DEFINITION L e t

(D,.)

o f groups. Denote b y ( t h e s e t ) D.

be a s e m i g r o u p a n d v a v a r i e t y (FD,v,t)

t h e f r e e group i n v on

6e "Free dgnr.'s"

181

F D ,V c o n s i s t s o f a l l f i n i t e sums C a i d i s where e q u a l i t y i s determined b y v . If e.q. 2)= then " e q u a l i t y " i s "formal equal i t y " . D e f i n i n g ( C o i d i ) - ( C a ! d ! ) : = ~ o i ( ~ a ! d . d ! )y i e l d s

7,

J J

6.21 THEOREM (a)

1

J

J

1

J

Let Z'Y be a v a r i e t y of g r o u p s .

i s well-defined.

( b ) ( F D , v , t , * ) = : F i s a n r . , d g . by D , whose a d d i t i v e group belongs t o v . ( c ) F o r every d g n r . (N',D') w i t h (",+I E 79 ever!! semiqroup homomorphism D + D ' can u n i a u e l y be e x t e n d e d t o a (F,D)-(N',D')-homomorphism.

( N , D ) with ( N , + ) homomorphic image o f ( F , D ) .

( d ) Every d g n r .

E

? i sl a (F,D)-(N,D)-

P r o o f . ( a ) : h o l d s by t h e d e f i n i t i o n of e q u a l i t y v i a laws in V . By a r o u t i n e b u t somewhat i a s t y c a l c u l a t i c n o n e sees t h a t ( F D , V , + , * ) i s a n e a r - r i n q . By c o n s t r u c t i o n , F 'is f r e e o v e r D i n 'V, s o ( F , t ) E z ) a n d D g e n e r a t e s (bj:

(F,+). ( c ) : By d e f i n i t i o n , e v e r y map

f:D

+

D'

can uniquely

h:(F,t) * ( N , t ) . If f i s moreover a s e m i q r o u p homomorphism, h i s a n (F,D)-(N',D')-homomorphism by 6 . 1 9 .

b e e x t e n d e d t o a homomorphism

( d j : C o n s i d e r i n g t h e d i a g r a m ( 1 i s t h e i n c l u s i o n map) a n d rememberinq o r o u p t h e o r y ( o r making a r o u t i n e d i a g r a m a r g u m e n t ) gives the information t h a t h i s a g r o u p - e p i m o r p h i s m . Now h / D = i d D , whence h i s a ( F . 0 ) - ( N , D ) - e p i m o r p h i sm by 6 . 1 9 See a l s o John ( l ) , Mahmood ( 1 ) - ( 4 ) , Meldrum ( 1 3 ) a n d Rhabari ( 1 ) , ( 2 ) . R e p r e s e n t a t i o n s o f groups v i a f r e e d g n r . ' s a r e s t u d i e d i n Meldrum ( 4 ) a n d ( 1 3 ) .

182

§ 6 0 lSTR IBUTIVE LY GENE R A T E D NEAR- R INGS

f ) D-GROUPS A N D ( N , D ) - G R O U P S

L i k e n r . homomorphisms o f d g n r . ' ~ , t h e c o n c e p t o f N - q r o u p s c a n be " r e f i n e d " f o r a d g n r . ( N , D ) : we w a n t t h e e l e m e n t s o f D t o " d i s t r i b u t e over r " ( t h i s appeared already in 6 . 5 ) . 6 . 2 2 DEFINITION Let (N,D) be a d g n r . an ( N , D ) - g r o u p i f V y l , y 2 ~ r

v

. N r ~ Ni qs dED:

called

d(ylty2) = dyl+dy2.

6 . 2 3 DEFINITION L e t ( D , . ) be a s e m i g r o u p and r i s c a l l e d a D-group i f a m u l t i p l i c a t i o n

(I',+) a :

a group.

r

Dxr-

(d,Y) dY VdcD: d ( y l t y 2 ) = d y l t d y 2 . +

i s defined with

y1,y2€r

(N,D) i s d g . t h e n N r i s a n ( N , D ) - g r o u p 6.24 REMARK So i f i f f r i s a D-group ( w . r . t . t h e r e s t r i c t e d m u l t i p l i c a t i o n of N r ) . Now l e t

(D,.)

by D i n 2, ''

be a s e m i q r o u p a n d F D , z ) be t h e " f r e e n r . d q . a s I n 6 . 2 1 , w h e r e 0 i s some J a r i e t y i n ? .

6 . 2 5 T H E O R E M E v e r y D-group Proof.

If

ZaidicF

D ,V

rEl)

i s an

and

ycr,

(FD,D,D)-group. define

(Coidi)y: =

= Cui(diy). Again t h i s i s well d e f i n e d and c h e c k i n g the (FD,Q,D)g r o u p a x i o m s c r e a t e s no p r o b l e m .

A g a i n , l e t q be a v a r i e t y o f g r o u p s a n d of a l l We c o n s i d e r t h e c l a s s (N,D) ( N , D ) -groups. L e t Q be t h e f a m i l y o f o p e r a t i o n s

.d

(+So,-)

u (Wn)nEN

(2,0,1)

u (')nCN

(N,D) rE'U

a dqnr. which a r e

of type

'

L e t % be t h e c l a s s o f ( u n i v e r s a l ) a l g e b r a s o f t h i s t y p e .

183

6f D-groups and (N,D)-groups

L e t q b e t h e v a r i e t y determined by a l l laws which d e f i n e ,

r&,

t o be a g r o u p

(I',t,O,-)

EV

for

and by a l l laws

Then c l e a r l y 6.26

THEOREM

3=(N,Dl,v(g;

so t h e l a t t e r c l a s s i s a v a r i e t y .

F r o m u n i v e r s a l a l g e b r a we now g e t

6.27

COROLLARIES T h e r e e x i s t a l l f r e e ( N , D ) - n r o u p s ; unique up t o the

6.28

(N,D)-isomorphisms;

(N,D)-(N,D)-homomorphic

each

they are

(N,D)-group

imaqe o f a f r e e

is

(N,D)-qroup.

REMARKS ( a ) Meldrum (2) used these " f r e e and t h e f r e e variety

9

faithful

(N,D)-groups

by

(N,Dj-group

i n

Zf"

(N,D)

has a

( n o t even i n the f i n i t e case). (even n o t every f i n i t e dgnr.)

i n s u c h a way t h a t

E(r)

r

remain d i s t r i b u t i v e on

endomorphisms on

D

i n a s u i t a b l e non-abelian

c a n b e e m b e d d e d i n t o some

dsD

dg.

t o show t h a t n o t e v e r y d q n r .

Therefore n o t every dgnr. a l l

nr.'s

( = become

r).

O b s e r v e t h a t we k n o w f r o m 6 . 1 1 t h a t e v e r y f i n i t e d a n r . c a n b e e m b e d d e d i n t o some that all

deD

E(r), i f

remain d i s t r i b u t i v e .

Meldrum a l s o c o n s t r u c t e d i n ( 2 )

( n , D ) , (N,D) such t h a t of

(N,D)

with faithful (N,D) and

one does n o t i n s i s t

i s a

(N,D)

"nearest" dgnr.'s

( m , D ) - ( (N,D)-)qroups

(R,D)-(N,D)-homomorphic is a

imaqe

-

(N,D)-(N,D)-homomorphic

(N,D). Moreover he considered t h e "Dorroh-type" image o f

of an i d e n t i t y 1 t o a drlnr. adjoin 1 t o

D)

(cf.

See a l s o E l e l d r u m ( 7 1 ,

(N,D)

(Kertgsz), (10)-(13).

Th.

adjunction

(one has t o

3.13).


184

( b ) F o r more i n f o r m a t i o n on

(N,D)-groups

see F r o h l i c h

(2). ( 4 ) . I n ( 4 ) . F r o h l i c h d e s c r i b e d f r e e sums a n d p r o d u c t s , o r t h o g o n a l sums, i n t h e case o f

f r e e bases and p r o j e c t i v i t y It turns out that the

(N,D)-groups.

s i t u a t i o n i s s i m i l a r t o t h e r i n g (-module)

case.

and ( N , D ) -

( c ) F r o h l i c h a l s o s t u d i e d c a t e g o r i e s o f N-

g r o u p s i n ( 5 ) and d e v e l o p e d a " n o n - a b e l i a n a l g e b r a " v i a these groups i n ( 6 )

-

homological

(8).

a \ S T R U C T U R E THEORY

We s t a r t w i t h a r e s u l t o n g e n e r a t o r s i n

6.29

N/I.

N and N

( N r ) b e fg.,

THEOREM ( ( G a s c h u t z ) ,

Lausch ( 4 ) ) .

N a dgnr. and I ( A )

be a f i n i t e i d e a l . Moreover,

N/I

(r/A)

Then

V

I;?

be t h e N-subgroup g e n e r a t e d b y

iEI1,

...,k )

t h e N-subgroup N

-Proof.

Let

3ei€Fi:

{el

,...,e,)

let

l,...,ek).

-

grznerates

(r).

As i n (Gaschiltz)

(where i t i s proved f s r groups;

t h i s p r o o f c a r r i e s o v e r t o groups w i t h o 3 e r a t o r s

-

see Lausch ( 3 ) ) . T h i s r e s u l t can be used t o p r o v e 6.31: 6.30

D E F I N I T I O N If N€ml

in

(N,*))

NEWl

I ( N ) : = {nENln i s i n v e r t i b l e

(N,.).

denotes t h e "group k e r n e l " o f

6 . 3 1 THEOREM ( L a u s c h ( 3 ) , Let

then

L a u s c h - N o b a u e r (l), Scott (1)).

be a f i n i t e dgnr. and l e t

homomorphism.

+)

h:N

+

m

be a nr.-

Then

+ ) T h e elements of I ( N )

a r e a l s o c a l l e d t h e " u n i t s " o f N.

185

6 g Structure theory

Proof.

(a) I f

iEI(N),

3

then

jEN:

= h(j)h(i) = h(l),

h(i)h(j)

( 0 ) Conversely,

if

t h e N-subgroup

h(N):

icI(h(N)) take

i j = j i = 1.

so

Hence

h(i)EI(h(N)).

Cil

then

generates

h ( n ) = Caih(di)ch(N) 1

and

3

= lukh(dk)

7.T = h ( 1 ) .

with

Then

11

= h(n).J.T

(iailuih(di)h(dk))i

= h(n).

1 1 1

S o b y 6 . 2 9 t h e r e i s some

h ( i ) = i and

with

is1

such t h a t t h e N-subgroup g e n e r a t e d by i e q u a l s N.-So t h e r e i s some

j E N

with

j i = 1.

Hence b y 1 . 1 1 3 ,

i i s invertible. 6.32

R E M A R K See L a u s c h ( 4 )

f o r some m o r e g e n e r a l v e r s i o n s o f

6.31. N e x t , we v i s i t p r i m i t i v e d q n r . ' ~ . w i t h D C C N and g e t 6.33

THEOREM ( L a r t o n ( 2 ) ) .

let

NcM(r)

be a f i n i t e d q .

non-

r i n g w i t h a l e f t i d e n t i t y . Then t h e f o l l o w i n g c o n d i t i o n s are equivalent: ( a ) N i s 1 - p r i m i t i v e on ( b ) N i s 2 - p r i m i t i v e on

r. r.

(c) N i s simple. ( d ) N = Mo(r) and m o r e o v e r

r

i s a finite,

non-abelian,

i n v a r i a n t l y simple group. Proof.

( a )

(d):

4.6(b),

r

( b )

( c ) f o l l o w s from 4.47(a).

N has an i d e n t i t y . By 4.60,

i s abelian,

i s a rina.

r . By M (r). GO

Assume t h a t N i s 2 - p r i m i t i v e o n N i s a b e l i a n b y 1.49,

Hence

r

i s non-abelian.

N

so by

i s an e n d o m o r p h i s m o f

r.

Since

all

dcNd,

whence

r

Nr

N

Since N i s f i n i t e ,

t h e same a p p l i e s t o r . r i s m o n o q e n i c , s o % 3 Y O E r : N / ( o : y o ) uN r by 3 . 4 ( e ) . So e v e r y

cannot c o n t a i n a n o n - t r i v i a l

If

6.6(c)

dsNd

i s N-simple,

it

subgroup i n v a r i a n t under

shows u p t o be i n v a r i a n t l y s i m p l e .

186

$ 6 DISTRIBUTIVELY GENERATED NEAR-RINGS

Now

i s f i n i t e and fixed-point-free, so i t

AutN(I')

c o n s i s t s e i t h e r of { i d } alone or contains a fixedp o i n t - f r e e a u t o m o r p h i s m o f p r i m e o r d e r . The p a p e r ( T h o m p s o n ) t e l l s us t h a t ( r , + ) i s n i l p o t e n t . B u t r i s invariantly simple a n d therefore abelian, a c o n t r a d i c t i o n . S o A u t N ( r ) = I i d l a n d M O(r) = Mo(r). c;

( d ) => ( b ) : I f I' by 4 . 5 2 ( b ) .

N = Mo(I')

t h e n F1 i s 2 - p r i m i t i v e o n

T h i s t h e o r e m h a s some i n t e r e s t i n o c o n c l u s i o n s ( s e e 7.46). We & o w c o l l e c t some r e s u l t s c o n c e r n i n g r a d i c a l s o f r e l a t e d d . g . n e a r - r i n g s . P r o o f s a n d more o n t h i s c a n be f o u n d i n K a a r l i ( 3 ) and ( 4 ) . 6 . 3 4 T H E O R E M ( K a a r l i ( 4 ) ) . L e t N be a d g n r , I a N and M SN N. (a) I f q&I i s quasiregular i n I then q is quasiregular i n N. 1 ( b ) J v ( M ) 3 J v ( N ) f l M f o r v = 0 and w = ~ . (c) If I , $ I 2 g . . . 4 I k a N and 1 4 1 , then

6.35 R E M A R K S ( a ) S u r p r i s i n g l y ( o r u n f o r t u n a t e l y ) , 6 . 6 ( a ) does not f o r c e

t h e v a r i o u s r a d i c a l s o f a d q n r . t o c o i n c i d e ( n o t even f o r f i n i t e d q n r . ' ~ ) . See s e v e r a l p a p e r s o f Laxton a n d Beidleman. Also, i s not necessary n i l i n t h i s c a s e . See a l s o S c o t t ( 1 1 ) .

22(N)

(b) For dgnr.'s

N E ? ~ ~ , Beidleman ( 8 ) d e f i n e d " s t r i c t l y

p r i m i t i v e " i d e a l s a s 2 - p r i m i t i v e maximal i d e a l s . The i n t e r s e c t i o n o f t h e s e o n e s c o n t a i n s C d 2 ( N ) and equals i n the c a s e o f D C C N ( t h i s f o l l o w s from 4.47(b)).

g2(N)

( c ) L a x t o n ( 3 ) c o n t a i n s a n e x a m p l e o f a f i n i t e d g n r . II w i t h

the property t h a t

g1,,2(N) i s

no i d e a l , w h i l e N h a s nilpotecr; l e f t ideals \ b u t o f course n o n i i p o t e n t i d e a l , .

187

6 g Structure theory

( d ) P e s k i n s ( 2 ) c o n t a i n s more i n f o r m a t i o n on t h e e N e ' s , w h e r e eEN i s some i d e m p o t e n t . (e) In (4), if

Tharmaratnam c a l l e s a dgnr.

EndN(N,+)

non-trivial instance.

l e f t ideals i s a division near-ring,

N - Mo(r).

for

N which i s not a

For a f i n i t e d i v i s i o n dgnr.

r i n g t h e r e i s some f i n i t e , with

N a d i v i s i o n dgnr.

A f i n i t e dgnr. without

AutN(N,+)u{6).

=

non-abelian

s i m p l e N-group

r

T h i s e s t a b l i s h e s a I - I - c o n n e c t i o n between

i s o m o r p h i s m c l a s s e s o f f i n i t e d i v i s i o n dg. ( f ) See T h a r m a r a t n a m ( 1 ) , ( 2 ) , ( 3 ) dgnr.'~": a topological nr. Beidleman-Cox ( 1 ) )

near-rings.

and ( 4 ) f o r " t o p o l o g i c a l N (def.

as u s u a l - s e e

i s c a l l e d a t o p o l o g i c a l dgnr.

if

Nd g e n e r a t e s N t o p o l o g i c a l l y .

If the topological nr.

N i s a dgnr.

then N i s a topo-

b u t t h e converse does n o t h o l d i n g e n e r a l .

l o g i c a l dgnr.,

Tharmaratnam a l s o d e s c r i b e d t o p o l o g i c a l

(N,D)-groups

and t h e s t r u c t u r e o f t o p o l o g i c a l d g n r . ' ~ , e s p e c i a l l y t h a t o f a 2 - p r i m i t i v e complete t o p o l o g i c a l dgnr.. ( 9 ) See L a x t o n ( 4 ) a n d L a x t o n - M a c h i n ( 1 )

f o r the behaviour

o f prime ideals i n d g n r . ' ~ . (h) Plotkin (1),(2)

transferes t h e concept o f a dgnr.

to

universal algebra. ( i ) See a l s o 9 7 c ) . ( j ) N i s c a l l e d a g e n e r a l i z e d dgnr. (gdg.nr.)

generates

(N,+).

nomial n r . ' s

D g n r . ' ~ , constant nr.'s

are of

by a l l g d g . n r . ' s

t h i s type.

3. E v e r y

is

i n a f i n i t e gdg.nr.. not a ring,

and " N

= M ( o ) ( r ) ' 'w i t h r

non-abelian

( k ) By 5 . 1 9 ( a ) a dgnr.

f i n i t e nr.

"I-primitive", finite,

are equivalent. and 6 . 3 4 ,

as a n i d e a l .

a n d many p o l y -

The v a r i e t y g e n e r a t e d

For f i n i t e gdg.nr.'s

and N o

i f Ndu Nc

N with i d e n t i t y

"2-primitive",

"simple"

i n v a r i a n t l y simple,

See P i l z - S o

n o t every N

c a n be embedded

€ a oc a n

(3). b e embedded i n

188

57

TRANSFORMATION NEAR-RINGS

T h i s c h a p t e r c o n t a i n s r e s u l t s on n e a r - r i n q s o f g r o u p mappings ( t h e "elements o f near-ring-theory''

o f 4.62)

r i n g s which a r e r e l a t e d t o these (97 d ) ) .

and o f n e a r -

We w i l l m a i n l y b e

concerned w i t h t h e i d e a l s t r u c t u r e o f t h e s e c l a s s e s o f n e a r rings

.

We s t a r t w i t h

fixed-point-free o f 2.50

where

M:(T)

i s shown t o f u l f i l l

i f f H h a s f i n i t e l y many o r b i t s o n

r.

We a l s o a n s w e r t h e q u e s t i o n , and

M!

r is

just H itself.

a l l conditions I n t h i s case,

M:(r)

i s s i m p l e and t h e f i n i t e t o p o l o g y on

Mi(T)

H i s some

group o f automorphisms of t h e ( a d d i t i v e l y

r.

w r i t t e n ) group

= (MH(r))o,

= MHU{61(l')

M:(T):

i s discrete.

M;

under which c o n d i t i o n s

(rl)

1

( r 2 ) are isomorphic, using semi-linear transformations 2 as i n r i n g t h e o r y . The a u t o m o r p h i s m g r o u p o f t h e M : ( r ) - q r o u p i n b ) we show t h a t f o r Mo(r)

Mo(I')

Turning t o

are equivalent:

a l l c o n d i t i o n s o f 2.50,

g e n e r a t i o n and f i n i t e n e s s o f

Mo(I')

a r e shown t o b e t h e

ones a r e a l l

(o:y)

(yer")

easy t o c h a r a c t e r i z e . (if

Ir(

2)

M(r)

r.

ACCL,

the followina

DCCL, f i n i t e

A l l minimal l e f t i d e a l s o f

(o:I*\{y?)

for

ycr*.

The m a x i m a l

a n d some o t h e r s , w h i c h a r e l e s s

C o n c e r n i n q i d e a l s we show t h a t are simple near-rinqs.

Mo(r)

and

There a r e no s u b n e a r -

Mo(r) a n d M ( r ) . I n c ) we s t u d y m a i n l y E ( r ) . E(r) i s 2 - p r i m i t i v e o n r i f f r i s i n v a r i a n t l y s i m p l e . I n t h i s case, E(r) = M o ( r ) . S i m i l a r A(r) a n d I ( r ) . E(r) h a s a l l results are obtained for

r i n g s s t r i c t l y between

c o n d i t i o n s o f 2.50 i f

r

i s t h e d i r e c t sum o f f i n i t e l y many

minimal f u l l y i n v a r i a n t subqroups. % i n v a r i a n t l y simple. Aut I ( r ) = r.

E(r)

i s simple i f f

r

i s

F i n a l l y we s t u d y n e a r - r i n q s o f p o l y n o m i a l s R[x] o r r[x] over a commutative r i n q R w i t h u n i t y o r a g r o u p r and t h e i r a s s o c i a t e d near-rings P(R), P ( r ) o f p o l y n o m i a l f u n c t i o n s . We show t h a t

7a Mfi(r)

iff R i s a finite field.

P ( R ) = f4(R)

i s simple i f f F i s i n f i n i t e . f u n c t i o n on F } ,

If

F =

If F i s f i n i t e but char

z2,

Z2,

but

and hence q u i t e w e l l - k n o w n ,

that

char F

2.

P(P)

i s simple i f f R i s a f i e l d

$; Z2

holds i f f

= M(T)

FLXJ

{ p ~ F [ x ] I p induces t h e zero

is a

e a c h i d e a l o f F[x]

Frx]

lrl>l.

F[x]

t h e r e a r e e x a c t l y 2 maximal i d e a l s .

F $. 2 ,

ring-ideal o f

-r [ x ]

If F i s a field,

If F i s finite,

c o n t a i n s e x a c t l y one maximal i d e a l :

iff

189

r

and

r[x]

B2

=

provided

or

r

i s simple i s a finite,

n o n - a b e l i a n s i m p l e g r o u p . We c o n t i n u e a n d c l o s e w i t h n r . ' s of

polynomials and polynomial f u n c t i o n s on R-groups.

Now we a r e g o i n g t o d e c o m p o s e

MH(r),

(r,+).

p o i n t - f r e e automorphism group o f I n c o n t r a s t t o 3.43,

w h e r e H i s some f i x e d -

we f i r s t d e c o m p o s e t h e i d e n t i t y a n d t h e n MH(r). B e f o r e d o i n q s o , we h a v e t o

get a decomposition o f f i x some n o t a t i o n .

7.1

approach i s i n Holcombe ( 4 ) .

A categorical

NOTATION T h r u u g h o u t t h i s s e c t i o n 7 a ) l e t

r

be a non-zero

group and H a f i x e d - p o i n t - f r e e automorphism qroup o f r . r = I o l u UBi b e a p a r t i t i o n o f r i n t o a d i s j o i n t ic1

Let

union o f orbits o f

r

under H.

Denote ( f o r i E I ) b y ei (4.28(a)) Of M H ~ { a( }r )

ei(Y) =

(SO

ei

t h e u n i q u e l y d e t e r m i n e d map with

Y

for

ycBi

o

for

y&Bi

i s l i k e the identity i n

M o r e o v e r , we a b b r e v i a t e s o m e t i m e s o r s i m p l y by M .

Bi

and o e l s e w h e r e ) .

MHJ(al

(r)

by

M;(r)

07

190

7.2


As promised,

THEOREM ( B e t s c h ( 7 ) ) .

(a) IeiIiEI) ( b ) A l l Mei

let

M.

=:

M:(T)

i s a s e t o f orthogonal idempotents.

n

=

Li

=:

(0:B.) j+i J

a r e l e f t i d e a l s and R-

groups o f type 2 w h i c h a r e M-isomorphic t o

r

and f u l f i l l

f o r Li.

ZFI TI

(c) M

Li.

i E I

(d) If

1 Li

L: =

=

I*Li iE1

i e I

then

f i n i t e l y many o r b i t s o n

r;

i f f H has

L = M:(r)

i n t h i s case,

1 ei.

1 =

i E I

(e) Every non-zero i n v a r i a n t subnear-rinq S o f M contains L. Proof.

( a ) i s e s t a b l i s h e d b y an e a s y c o m p u t a t i o n .

4.28(aj,

Conversely i f

T h e map

[I ( o : B j ) . j+i 11 ( o : B .J) t t i e n j+i

Mei 5

whence m

E

f y : Mei

+

mei

+

r

m = me.EMei, 1

i s an M-epimorphisrn f o r

meiy

y€Bi.

Ker f Ker f

Y

= Mei n ( 0 : ~ ) . B u t

fl

=

(o:Bj)

=

IO),

(o:y)

and

fY

j E I

b e an M - i s o m o r p h i s m f r o m i s 2 - p r i m i t i v e on

to

L~

?r PM

r

Li

by 4.52(b),

= (o:Bi),

to

r.

so

i s unmasked t o Since

M:(r)

t h e same a p p l i e s

r.

( c ) i s s e t t l e d by t h e M-isomorphism s e n d i n q m i n t o (...,mei,...).

f:

M:(r)

+

Jl Li is1

191

7a M a ( r )

(d)

n

Li =

ieI

8

Li

h o l d s i f f I i s f i n i t e . Now a p p l y

ic1

2.30.

( e ) I f i e I a n d S as d e s c r i b e d , ( o : L i ) = ( o : r ) by 1 . 4 5 ( b ) . S u p p o s e t h a t Lin S = ( 0 1 . Then S L i c L i n S = = {Ol, s o S F ( o : L i ) = ( o : r ) = l o ) , a c o n t r a d i c t i o n . So L i n S { O l , whence L i n S = L i by ( b ) , s o a l l L j c S a n d t h e r e f o r e LsS.

+

R a m a k o t a i a h ( 7 ) showed t h a t t h e L i ' s a r e e x a c t l y a l l m i n i m a l l e f t i d e a l s o f M:(r). He a l s o c h a r a c t e r i z e d i n t h i s p a p e r a l l maximal l e f t i d e a l s ( a l s o i n terms o f t h e f i n i t e t o p o l o a y i n 4.26). The f o l l o w i n g r e s u l t g e n e r a l i z e s Theorem 5 . 7 o f B e t s c h ( 7 ) ( n o t a t i o n a s a b o v e ) . C f . Ramakotaiah ( 3 ) . 7.3

C O R O L L A R Y The f o l l o w i n q s t a t e m e n t s s r e e q u i v a l e n t :

(a) M = L. ( b ) MM f u l f i l l s a l l c o n d i t i o n s o f 2 . 5 0 . ( c ) H h a s f i n i t e l y many o r b i t s on r . Proof: apply 7.2. 7.4

C O R O L L A R Y I f M f u l f i l l s the c o n d i t i o n s o f 7 . 3 t h e n M has no n o n - t r i v i a l t w o - s i d e d i n v a r i a n t s u b n e a r - r i n g s . I n p a r t i c u l a r , M i s simple.

T h i s f o l l o w s from 7 . 2 ( e ) ( s i m p l i c i t y can a l s o be d e r i v e d from 4 . 4 6 ) . More on s i m p l i c i t y o f M:(r) c a n b e f o u n d i n Meldrum (12).

7.5

C O R O L L A R Y ( B e t s c h ( 7 ) ) . Let t h e n o n - r i n q N E f l o n % , be 2 - p r i m i t i v e on r . T h e n t h e f o l l o w i n a c o n d i t i o n s a r e equivalent:

( a ) NN i s " f i n i t e l y completely reducible" ( a l l conditions of 2.50 are v a l i d ) . and G h a s f i n i t e l y many o r b i t s on r . ( b ) N 1 M:(r) ( c ) N 2 ll;(r)

a n d t h e f i n i t e t o p o l o n y on

Mi(r)

i s discrete.

07

192


Proof. 2 . 5 0 , 4.60, 7.3 a n d a.29. There i s an i n t i m a t e c o n n e c t i o n between t h e l a t t i c e s a l l H - i n v a r i a n t s u b g r o u p s o f r and f(M) = ( S S M I S M S S I " r i g h t - i n v a r i a n t su b n e a r - r i nqs " o f 11) We mention w i t h o u t p r o o f :

.

7 .6

yH(r) o f (the

T H E O R E M ( L a x t o n ( 2 ) , Be t sc h ( 7 ) , 9 8 ) . I f H has S E I N o r b i t s on r* t h e n t h e map f : g H ( r ) + Y(M) i s a l a t t i c e isomor A

phism w i t h Moreover,

+

(A:r)

+ q H ( r ) given

f-l: f(M)

[ ( j H ( r ) =l

\!f(M)I

S

by

S

+

sr.

2'.

Holcombe ( 6 ) s u g g e s t e d t h e f o l l o w i n g

7.7

DEFINITION Every c h o i c e B : = { b i l i ~ I ) o f r e p r e s e n t a t i v e s i s c a l l e d a H-base. dirnH(r): = I B I i s c a l l e d t h e H-dimension o f r . biEBi

T h i s comes from t h e e a s y - t o - p r o v e

7.8

( c f . 4.28)

PROPOSITION (Holcornbe ( 5 ) ) .

(a)

ycr*

( b ) Each map

3

3

icf B

+

r

hcH: y = h ( b i ) .

can be u n i q u e l y e x t e n d e d t o a map

MH(r)m Holcombe f o r m u l a t e d 7 . B ( b ) more g e n e r a l l y : "Every m a p B + I", where r ' i s a n o t h e r g r o u p o n which H o p e r a t e s ( r ' i s a n "H-group") c a n be e x t e n d e d t o a u n i q u e map r * f ' which commutes w i t h H " . So r i s i n a k i n d " f r e e " o n B . H o p e r a t e s on $ ( r ) i n a n a t u r a l way. From 7 . 8 ( b ) we g e t 7 .9

T H E O R E M (Holcombe ( 5 ) ) . I f

d i m H ( M H ( r ) ) = (S+l)'-l.

d i r n H ( r ) = SEIN t h e n

7a M;(r)

7.10 REMARK I n ( 7 ) , pp. 92-97,

Betsch studied t h e d i s t r i b u t i v e

0

D:

elements

193

= (MH(T))d

bl:(r)

of

and "monomial m a t r i c e s "

matrices over

D

which c o n t a i n i n each column

a t most one non-zero e n t r y

-

cf.

over D (i.e.

i f H h a s f i n i t e l y many o r b i t s o n

r.

isomorphic t o t h e monoid o f a l l

fcEnd(r)

with all

haH.

D i s

also Frohlich (3)).

shown t o b e embeddable i n t o t h e s e m i q r o u p

(End (M,+),o)

(D,.)

i s antiw h i c h commute

Deskins (2).

Cf.

M;

Now we c o n s i d e r t h e f o l l o w i n g p r o b l e m : w h e n a r e

(rl)

and

1

(r2) isomorphic ? 2 F o r r i n g s , t h i s p r o b l e m i s s o l v e d i n t h e f G l l o w i n o way ( s e e

M:

i s an homomorphism rings

(V1,V2

v e c t o r spaces o v e r t h e d i v i s i o n r i n o

r e s p e c t i v e l y ) t h e n h i s an isomorphism i f f t h e r e

D1,D2,

i s some 1 - 1 - s e m i - l i n e a r

!+EHomD ( V l , V l ) :

t: V 1

transformation

V2

+

such t h a t

h(4) = t 0 t - l .

1

We f o l l o w i n some w a y J a c o b s o n ' s d i s c u s s i o n a n d s t a r t w i t h

7.11 THEOREM ( H o l c o m b e (4), R a m a k o t a i a h ( 6 ) ) . (as usual) fixed-point-free

r

M:l(r)

then

(r)

= M;

H1EH2

If

are

groups o f automorphisns on H1 = H E .

2 P r o o f . We o n l y h a v e t o s h o w "->". Suppose t h a t

H1=H2,

and c o n s i d e r t h e o r b i t s

ycr*

with respect t o Clearly

3

a n d t a k e some

H1,H2.

but

h2(y)€B2,

Then

fixed-point-free, 4.28(a)

0

B2 = H2y.

and

(since otherwise

hl

= h2

since

g u a r a n t e e s t h e e x i s t e n c e o f some

ml(h2(Y)) But

containino y

B1 = H1y

so

Take

H2

i s

a contradiction).

ml(y) = h 2 ( y ) '

with

Bl,B2

h2(y)BBl

~ ~ E H ~ Ch l (Hy ) ~ := h 2 ( y ) ,

h2cH1\H2.

and

6BB1:

ml(6) =

mlcM:l(r) Hence

0.

= 0-

= ml(h2(~))

= h2(ml(y))

m EM' ( r ) = M i ( r ) . T h u s H1 2 a c o n t r a d i c t i o n . H e n c e HI

= h2(h2(y))

h2(y) = H2.

0,

since

whence

y =

0,

$ 7 TRANSFORMATION NEAR-RINGS

194

MH1(r)

7.12 REMARK Observe t h a t

= MH2(r)

(r)

M:

= M:2(r).

1 7 . 1 3 COROLLARY ( R a m a k o t a i a h ( 5 ) ) .

(r)

Aut

M;P) Proof.

H':

If

=

AUtMH(r)

(r)

then

H = H'

point-free

on

7.14 DEFINITION

M:2(r2)y

M:

ri

SEHom(rl,r2)

homomorphism

MH(r)= MH,(r).

b y 7.11.

( r l ) and 1 (i = ly2).

N e x t we c o n s i d e r

H I i s by 4.52

HEH'.

shown t o b e f i x e d - p o i n t - f r e e w i t h SO

= H.

if

3

w h e r e Hi

are fixed-

i s called a semi-linear H1*

s:

W

H2

yl€.T1

S(hl(Y1)) = S(hl)(S(Y1)).

If

S

+

tl

6,

hlEH1:

s i s uniquely

d e t e r m i n e d and c a l l ed t h e isomorphism associated w i t h

-S .

w i l l a l s o speak a b o u t t h e s e m i - l i n e a r monomorphism

( Y ).

We

(S,S).

7 . 1 5 T H E O R E M ( R a m a k o t a i a h (5), f o r t h e f i n i t e - d i m e n s i o n a l

(7.7)

case a l s o B e t s c h ( 7 ) and Holcombe ( 5 ) ) . homomorphism

0

(rl)

f : PIH

+

tl;

A near-rino i s a n i s o m o r p h i s m

(r2)

1 2 t h e r e i s some s e m i - l i n e a r i s o m o r ! J h i s m mcMil(r,): f(m) = Som0S-l.

w

Proof.

We a b b r e v i a t e

M:

(ri)

by

Mi

S:r,1

(iE{1,2)),

+

r2

with

and

i keep t h i s n o t a t i o n f o r i. :

Assume now t h a t

(a)

M1

f : M1

M2

+

i s an isomorphism.

can be c o n s i d e r e d t o b e 2 - p r i m i t i v e on

M1Xr2

since

+

+

(“19Y2)

+

f ( m 1 )Y 2

( b ) We s h o w t h a t t h e r e i s a n i s o m o r p h i s m -1 mlEM1: f(ml) = SomloS

M1

By 7 . 2 ( b ) ,

say).

rl

So b y ( a )

= ml(S(yl))

and

W

rl r2 W

s(ml(vl))

mlEM1:

whence

= f(ml)(S(yl)),

r2

By ( a ) ,

4.56(a)

a r e M1-isomorphic

Ylcrl

( c ) Now we c l a i m t h a t

hl

r2.

and o n

-c

Soml

(by S , =

= f(ml)oS

= Somlo~-l.

f(ml)

Clearly

r1

contains a minimal l e f t ideal.


assures t h a t

or

S:

.

with

M1

r2,

does t h e r e q u i r e d j o b .

r2

S0hloS-’cEnd(

commutes w i t h a l l

( b ) and 7.13

hltzH1:

SohloS-1cH2.

r2). I f f ( m l ) = r n 2 we g e t f r o m

mlcM1.

m20(Snh,oS-’)

=

f(ml)o(SohloS-’)

=

= S a m , ~ S - ’ ~ S ~ h l o S - ’= S 0 r n l ~ h l ~ S - l = SohlomloS - 1 = ( S O ~ ~ O S - ’ Henc’e ) ~ ~ ~S o. h l 0 S - ’

( d ) N e x t we o b s e r v e t h a t isomorphism, procedures.

s : H1 hl

EAutM

+

+

2

(r)

H2

=

= ti2.

i s an

SohloS-’

a f a c t which can be seen b y t h e usual

196

§ 7 TRANSFORMATION NEAR-RINGS

( e ) F i n a l l y , we h a v e t o c h e c k t h e s e m i - l i n e a r i t y c o n d i t i o n 7.14

some

hlEH1

for

t a k e some

(S,s):

.

= AutMl(r)

= (Soh1oS-l) (S(Y1) 1

Then

S(hl(yl))

= s ( h 1 ) (S(Y1)

and

yl~rl

=

1.

The p r o o f i s now c o m p l e t e . From t h a t we c a n d e d u c e i n t e r e s t i n g r e s u l t s a b o u t t h e automorphism o f n e a r - r i n g s o f t h e t y p e 7.16

COROLLARY ( R a m a k o t a i a h ( 5 ) ) . then S on

fEAut

r

M

with

M:(T):

M = Mi(r)

If

and

fEEnd(M)

t h e r e e x i s t s a s e m i - l i n e a r automorphism

f ( m ) = SomoS’l

for all

mcM.

M 2 = M.

M1

T h i s f o l l o w s f r o m 7.15 b y s p e c i a l i z i n q

7.17 THEOREM ( R a m a k o t a i a h ( 5 ) ) . L e t G b e t h e p r o u p o f s e m i l i n e a r a u t o m o r p h i s m s o n r a n d G ‘ : = Aut M:(l‘). % Then G / G n H G’,

-

Proof. Define some

a:

as f o l l o w s :

G’

-*

M:(r)

fEAut

(by 7.16).

G

with

if

there i s

SEG, f(m)

mEMi(r):

= SornoS-’

Observe t h a t t h i s f i s unique. P u t

a ( S ) : = f.

F i r s t we p r o v e t h a t a i s a homomorphism. take

S,TEG.

Then f o r a l l

a(ST) = :

g,

a ( S ) =:

mEM:(r)

q(m)

fl,

= (ST)m(ST)-l

= S T ~ T - ~ S= - s~ f 2 ( m ) S - l = f 1( f 2( m 1 )

Hence

To do t h i s , a ( T ) =:

g = flf2# i m p l y i n g t h a t

f2.

=

.

a(ST) = a ( S ) a ( T ) .

Now a i s a n e p i m o r p h i s m : i f f E G ’ then there i s some S E G w i t h b mcM:fr): SornoS-’ = f f m ) . Thus a(S) = f.

F i n a l l y we c o m p u t e K e r a . I f SEKer a t h e n b m E M i ( r ) : m = i d ( m ) = SmS”. So

SEAut

$(r)

(I-)

Conversely,each T h i s shows t h a t

= H

(7.13).

element o f

6/Gn

%

G n H G’.

Hence is i n

SEGnH.

Key a.

7b M ( r ) and M,(r)

197

U is C l a y ( 1 4 ) d e t e r m i n e d the g r o u p U o f u n i t s o f ME(r): isomorphic t o the wreath p r o d u c t o f G w i t h t h e symmetric group u n d e r G . He a l s o p o i n t e d on t h e i n d e x s e t I o f t h e o r b i t s o f o u t t h e i n t i m a t e c o n n e c t i o n b e t w e e n U and t h e g e n e r a l l i n e a r g r o u p s i n l i n e a r a l g e b r a . He a l s o d e f i n e d a " d e t e r m i n a n t

f u n c t i o n " on U. S e e a l s o p . 3 7 6 . I n ( 3 ) , R a m a k o t a i a h showed t h a t M i ( T ) i s a r i n g i f f dimensional vector space over t h e skew-field H .

r

is a 1-

M:(r) f o r H s E n d ( r ) i s s t u d i e d i n 99 h ) . Don't f o r g e t t o read t h i s c h a n t e r a s we1 1 !

b)

M,(r)

AND

M(T)

T h e r e a r e a l o t o f t h i n g s w h i c h we c a n g e t by s p e c i a l i z i n g H = { i d ) i n t h e p r e v i o u s s e c t i o n . By 1 . 1 3 i t i s j u s t i f i e d t o consider primarily Mo(r). We s t a r t by c o n s i d e r i n g l e f t i d e a l s i n M o ( r ) . C f . Holcombe ( 4 ) . 7.18 C O R O L L A R Y ( H e a t h e r l y ( l ) , ( 4 ) , Ramakotaiah ( 7 ) ) .

(a) If for then

dcr,

e

{e616Er*)

6

:r

+

r

cf. a l s o Frohlich ( 3 ) ,

with

{

ed(y) =

d

if

y = 6

o if yS.6 i s a s e t of orthogonal idempotents.

( b ) A l l Mo(r)e6 =: L6 = (o:r\C6) a r e l e f t i d e a l s and M,(r)-groups o f t y p e 2 ( h e n c e minimal M o ( r ) - s u b g r o u p s ) g e n e r a t e d by e d a n d M o ( r ) - i s o m o r p h i c t o r.

If

then

iff

r

is finite.

( e ) Every non-zero i n v a r i a n t s u b n e a r - r i n a S o f c o n t a i n s L.

Proof: 7.2.

Mo(r)

§ 7 TRANSFORMATION NEAR-RINGS

198

Hence Mo(r) i s a 2 - p r i m i t i v e n r . o n T w i t h i d e n t i t y a n d a minimal l e f t i d e a l ( s e e 5 4 d 3 ) ) . 7 . 3 and 7 . 1 8 g i v e 7.19 C O R O L L A R Y ( H e a t h e r l y ( 3 ) , M . Johnson ( 6 ) ) . The f o l l o w i n g are equivalent: (a)

Mo(r)

1'

=

L

= L.

6E r *

(b) Mo(r)

has D C C L .

( c ) Mo(r) h a s A C C L . (d)

Mo(r)

i s completely r e d u c i b l e i n t o f i n i t e l y many

minimal l e f t i d e a l s .

( e ) Mo(r) (f)

Mo(r)

(9)

r

has o n l y f . g .

l e f t ideals.

i s finite.

is finite.

Mo(r)

C l e a r l y (by 7 . 4 )

i s s i m p l e i n t h i s c a s e . However, we

w i l l extend t h i s r e s u l t t o t h e a r b i t r a r y c a s e ( 7 . 3 0 ) . B u t f i r s t we examine t h e l e f t i d e a l s more c l o s e l y ,

7 . 2 0 T H E O R E M ( H e a t h e r l y ( 3 ) ) L e t L be a l e f t i d e a l of (a)

V

( b ) If

A:

= I 6 ~ r l L 6=

Proof. ( a ) I f

b

or

YEr: L y = I01

Ly

+

I01

rl

Ly = -/=

d

then

mcMo(r): m R E L , whence f o r t i o r i Ly = r .

Mo(r).

r. then

3

EEL:

1L

6 ~ E A Ey

5

L.

4 0.

ImLyJmcMo(r)) =

But

r

and a

( b ) I t s u f f i c e s t o show t h a t i f L6 4 I o l t h e n L6%L. This t r S v i a l l y holds f o r l r l r 2 s i n c e then l M o ( ~ ) 1 ~ 2 .So assume t h a t 11-123.

Suppose t h a t k 6 ~ L 6 ( ~ E A ) . Denote E , ( 6 ) = : 8 . Choose some I1cL w i t h t ( 6 ) = 6 = E 6 ( 6 ) . This i s p o s s i b l e by ( a ) . e y = e Take m.nEMo(r) with m(y) = 0

YS;O

199

7b M ( r ) and M,(r)

(so

=t 6

Y

m = ee

d h e -

n(y)

1: =

Then

-

of 7 . 1 8 ( a ) ) ,

and

b u t for

n(6) = 0,

L(Y)>.

and for

m(n+l)-mnEL

y =/= 6

we q e t

t(v)

= m(n(y)+k(v))-m(n(v)) 0-0 = 0 L6(y), w h i l e T(S) = m ( n ( a ) t ! L ( & ) ) - m ( n ( 6 ) ) = m ( e ) - m ( o ) =

e

=

=

So

e,(s).

-

l6 = R E L

a n d we a r e t h r o u g h .

7 . 2 1 COROLLARIES ( B l a c k e t t ( I ) ,

( a ) The L 6 ' s of M o ( r ) .

(6Er*)

H e a t h e r l y ( 3 ) , PI. J o h n s o n ( 1 ) , ( 3 ) ) .

a r e e x a c t l y a l l minimal l e f t i d e a l s

( b ) Every l e f t i d e a l of Mo(r) which i s c o n t a i n e d i n L 1 L i s i s o m o r p h i c t o a d i r e c t sum o f s u i t a b l e 6Er*

L * k ( c ) L c a n n o t be a n o n - t r i v i a l d i r e c t summand o f Proof. -

Wo(r).

( a ) i s a c onse que nc e of 7 . 2 0 .

( b ) f o l l o w s from ( a ) and 2 . 5 5 . ( 6 )

If

L i L ' = Mo(r)

L' dL Mo(r) i s suc h t h a t

L ' $. {Ol t h e n ( b y 7 . 2 q ) t r a d i c t i on.

3

6Er*:

Lg

5 L n L ' ,

and

a con-

S i n c e we have been v e r y s u c c e s s f u l i n d e t e r m i n i n g a l l minimal l e f t i d e a l s o f Mo(r), we t u r n t o maximal l e f t i d e a l s . ble r e a d i l y g e t some of them: 7 . 2 2 E X A M P L E ( H e a t h e r l y ( l ) , ( 3 ) ) . For e v e r y

i s a maximal Mo(r)-subgroup i d e a l ) o f Mo(r).

-Proof.

By 3 . 4 ( e ) a n d 7 . 2 we qe? N / ( o : v )

where

N =

y~r*,

(o:y)

( h e n c e a l s o a maximal l e f t

Mo(r). Now a p p l y 3 . 4 ( h ) .

T

=N

Lyy

07

200


B u t woe: 7.23

PROPOSITION ( H e a t h e r l y ( 3 ) ) .

If

r

i s i n f i n i t e then L

i s c o n t a i n e d i n a maximal l e f t i d e a l o f Mo(r), b u t n o t i n a n y ( o : y ) (yEr*). (as i n 7.18(d))

-

P r o o f . The f i r s t s t a t e m e n t i s s e t t l e d b y 1 . 5 3 ( a )

Mo(r)cgl Lyc(o:y)

since

a n d b y 7 . 1 8 ( d ) . Now i f L ~ ( o : y ) t h e n a n d e y ( y ) = 0 , w h e n c e y = 0.

So t h e r e a r e o t h e r m a x i m a l l e f t i d e a l s b e s i d e t h e

(o:y)'s,

which i m p l i e s more t r o u b l e f o r us. But, f o r t u n a t e l y ,

M.

Johnson

h a s s o l v e d t h i s p r o b l e m . See a l s o R a m a k o t a i a h ( 7 ) a n d ( 8 ) .

7 . 2 4 NOTATION F o r

mEMo(r)

call

CyErlm(y)

= 0 1 = : Z,

(the

"zero s e t o f m"). We s t a t e w i t h o u t p r o o f 7.25

PROPOSITION ( M . J o h n s o n ( 3 ) , ( 6 ) ) . i n Mo(r). T h e n

( a ) EEL (b) 7.26

(o:ZE)eL.

3

al,L2cL

NOTATION L e t

= IL

L e t L be a l e f t i d e a l

E E L : ZEln Z E 2 = Z E .

$(r)

b e f o r t h e moment

=

x

=

AE Mo(r)l \ & E L : ZL i s i n f i n i t e ] .

Now we a r e i n a p o s i t i o n t o c h a r a c t e r i z e t h e m a x i m a l l e f t ideals o f 7.27

Mo(r).

THEOREt4 ( ? 4 . J o h n s o n ( 6 ) ) . L e t L b e a ? e f t i d e a l o f M o ( r ) . L i s m a x i m a l ( 3 y E r + : L = ( 0 : ~ ) )v ( L i s m a x i m a l i n

t(r)). P r o o f . =>: that

Suppose t h a t yEr: L ( 0 : ~ ) . Assume m o r e o v e r E E L : Z1, i s f i n i t e . F o r YE:T c o n s i d e r e, of

3

7.18(a).

S i n c e L i s no

(o:y),

all

eyEL

b y 7.20(b).

7b

M(r)and M,(r)

201

Assume t h a t l z t l = n E N . We c l a i m t h a t 1 kcL: z k ( 0 1 a n d p r o v e t h i s by i n d u c t i o n on n. ( a ) T h i s i s t r i v i a l f o r n = 1. ( b ) Suppose t h a t n > l and t h e s t a t e m e n t h o l d s f o r n-1. Now t + e E L and Zll+e = Z,\{y1, so Y Y IZllte 1 = n-1. Y S o L = Mo(r) by 7 . 2 5 ( a ) s i n c e ( o : Z k ) = ( o : o ) = LEL(r) = Mo(r)cL. T h i s i s a c o n t r a d i c t i o n . Hence a n d s i n c e L i s a maximal l e f t i d e a l , L i s maximal i n $.(r).

=

I n v i e w o f 7 . 2 2 i t s u f f i c e s t o c o n s i d e r niaximal e l e m e n t s o f $. L e t K qll Mo(r) p r o p e r l y c o n t a i n L . T h e n 3 kcK: l Z k l = n e N . A g a i n we use i n d u c t i o n t o show t h e e x i s t e n c e o f some klcK w i t h = {ol, zkl n = 1 is ti-ivial again. 0

fp]

= 1.

[p]

also holds. p~R[x]

with

i s r i g h t cancellable.

'

1 aiq i

poq =

= a n b m x n m t summands o f l o w e r d e g r e e .

i= O

(b) follows from (a).

( c ) If

f o p = qop

= 0,

[f-q] and so

f-q

q o p = fop

7.69 DEFINITION C a l l

[f]

implies

so

p~R[x]

= [p]

then

or

( f - q ) o p = 0.

i s a constant c. qop+c,

whence

(a)

[p]cP

(b)

If p1

[p]

f = q+c

p = fog

= [p].

7 . 7 0 PROPOSITION ( L a u s c h - N o b a u e r ( 1 ) ) . domain and

Hence

c = 0.

indecomposable i f [g]

By ( a ) ,

L e t R b e an i n t e q r a l

p~R[x].

->

p i s indecomposable.

> 1

then t h e r e e x i s t indecomposable polynomials

,. . . , p s ~ R [ x ]

with

p = p10p20.. .ops.

7d Polynomial near-rings

Prosf. (a) i s t r i v i a l . ( b ) follows by i n d u c t i o n on

21 9

usinq 7,68(a).

[p!,

7 . 7 1 DEFINITION ( a ) L e t p be c a l l e d normed i f

a n = 1.

(b) p i s called linear i f

= 1.

[p]

Now we can s h a r p e n 7 . 7 O ( b ) a n d s t a t e w i t h o u t p r o o f : 7 . 7 2 T H E O R E M ( L a u s c h - N o b a u e r ( 1 ) ) . I f F! i s a f i e l d o f c h a r a c t e -

p~R[x]

r i s t i c 0 and i f

has

[p]

1

then

~ ? o p ~ o . . . o pw ~ i t h II l i n e a r , (a) p indecomposable and normed.

(b)

If

p = moqlo

of p then and q j ' s

...Oqt

S E N and

pl,

...,p,

i s a n o t h e r such "prime decomposition"

m = II, s = t a n d t h e d e g r e e s o f t h e p i ' s a r e t h e same ( u p t o o r d e r ) .

( c ) There e x i s t o n l y f i n i t e l y many d e c o m p o s i t i o n s o f

p of

t h e form ( a ) . T h i s u n i q u e n e s s can even be s t r e n q t h e n e d by t h e deep theorem 2.46 o f Lausch-Nobauer ( 1 ) . 7.73 R E M A R K See Graves-Malone ( 2 ) f o r a n o t h e r r e s u l t of "prime f a c t o r decompositions" in "Euclidean near-domains'' ( s e e 9 . 6 0 and 9 . 6 7 ( c ) ) .

7 . 7 4 DEFINITION C a l l

REQ p o l y n o m i a l l y c o m p l e t e ( N o b a u e r ( 6 ) )

o r 1 - p o l y n o m i a l l y complete (Lausch-Nobauer ( 1 ) ) if P ( 2 ) = M ( R ) , i . e . i f e a c h map R R i s a ?olynomial -f

function.

87

220


O f c o u r s e , Z , I, IR a n d (c a r e n o t p o l y n o m i a l l y c o m p l e t e . O n t h e o t h e r h a n d , t h e f a c t t h a t q, i s p o l y n o m i a l l y c o m p l e t e i s of high v a l u e i n mathematical l o g i c . P e r h a p s we now f e e l w h i c h R m i g h t be p o l y n o m i a l l y c o m p l e t e and prove the f o l l o w i n g i n t e r e s t i n o r e s u l t , w h i c h w i l l a l s o prove useful i n the sequel. 7 . 7 5 T H E O R E M ( N o b a u e r (6), L a u s c h - N o b a u e r

R i s p o l y n o m i a l l y c o m p l e t e

(1)). R i s f i n i t e and simple

(hence a f i n i t e f i e l d ) . P r o o f . ->: I f R i s i n f i n i t e , [ P ( R ) I S IR[x]I = IRI < so R c a n n o t b e p o l y n o m i a l l y < IRIIRI = IM(R)I, complete. Let I b e a n o n - t r i v i a l i d e a l o f R . T a k e some nonzero iEI. I f ~ E P ( R ) then p ( i ) - p ( O ) c I , since I SIR. Let f c M ( R ) f u l f i l l f ( i ) - f ( O ) e I . Then f t P ( R ) and a g a i n R i s n o t polynomially complete. T h i s f o l l o w s from L a g r a n y e ' s i n t e r p o l a t i o n theorem ( o r f r o m t h e f a c t t h a t P ( R ) i s 2 - p r i m i t i v e

: = ( L J : R ) ~ C ~ I4 R C X ] .

show u p t o be a l s o These examples of i d e a l s i n ( R [ x ] , t , o ) ideals in (R[x],+,*). This l e a d s t o t h e followinrl i s c a l l e d a f u l l i d e a l (Nobauer) o r 7.82 D E F I N I TION I 9 REX] T - 0 - i d ea l (Menger) o r c o m p o s i t i o n i d e a l ( A d l e r ) i f a l s o

I 4 (R[x],+,*). 7 . 8 3 R E M A R K Thus f u l l i d e a l s a r e e x a c t l y t h e " i d e a l s " o f t h e c o m p o s i t i o n r i n g ( T O - A l p e b r a ) ( s e e 1 . 1 1 7 ) (R[x] , t , * , o ) . a r e much b e t t e r e x p l o r e d In f a c t , t h e s e f u l l i d e a l s of R[x] t h a n t h e " o r d i n a r y " i d e a l s o f R[x]. Not e v e r y i d e a l o f R[x] i s a f u l l i d e a l , b u t t h i s holds in s p e c i a l , important cases (see 7.93). I f e . g . R happens t o be a f i e l d t h e n a l l f u l l i d e a l s a r e p r i n c i p a l ( a well-known r e s u l t o f r i n q t h e o r y ) a n d so q u i t e e a s y t o o v e r l o o k . I t i s n o t k n o w n u n d e r which c o n d i t i o n s each i d e a l o f R[x] i s " p r i n c i p a l " ( = g e n e r a t e d by one s i n g l e polynomial), or a t l e a s t f . g . , The i d e a l s of 7 . 8 1 show u p t o be q u i t e i m p o r t a n t o n e s : 7.84 T H E O R E M (Nobauer ( l ) , Lausch-Nobauer ( l ) , Hule ( 1 ) ) . L e t I be a f u l l i d e a l o f R t x ) . Then t h e r e e x i s t s e x a c t l y one i d e a l J of R w i t h ( J ) ' I I " < J > (namely J = I n R ) . J : = I nR. Then c l e a r l y ( J ) = J [ x ] e I . Also, i f iE1 a n d r E R t h e n i ( r ) = i o r E I n R = J , whence I c < J > . I f J ' 9 R a l s o f u l f i l l s (J')sI'<J'> then ( J ' ) c < J > , whence J ' e < J > a n d s o J ' c J . S i m i l a r l y , J e J ' and hence J = J ' .

P r o o f . Let

223


7 . 8 5 DEFINITION- L e t I , J be a s i n 7 . 8 4 . Then J i s c a l l e d t h e " e n c l o s i n q i d e a l " o f I ( J i s n o t i d e a l i n R[x] in general ! ) .

F o r (much) more i n f o r m a t i o n c o n c e r n i n g t h e s e e n c l o s i n g i d e a l s s e e t h e c o m p r e h e n s i v e book L a u s c h - N o b a u e r ( 1 ) . C f . a l s o Mlitr (1).

We w i l l g e t more p o w e r f u l r e s u l t s when

?i

i s a s s u p e d t o be

a field:

5.) F r x l Throughout t h i s number, l e t F d e n o t e a commutative f i e l d . 7 . 8 6 PROPOSITION ( C l a y - D o i ( Z ) , S t r a u s ( 1 ) ) . w i t h L n F =/= I 0 1 ( c f . 7 . 7 8 ( b ) ) . (a) F

E

(b) If

Let

L AII F[x]

L. IF1 > 2

then

L = F[x].

P r o o f . ( a ) L e t II f 0 be c L n F . By 7 . 7 9 ( b ) , f . E - l . 9 . = fEL.

Then t a k e some f e F .

By ( a ) , fEL ( b ) I f c h a r F $. 2 , t a k e f : = 2 - ' . 2 2 2 and a l s o f E L . Hence x o ( x + f ) - x O X E L , so x + f 2E L , w h e n c e X E L . Use ( a ) a n d a p p l y 7 . 7 9 ( c ) t o get L = I f char F = 2 , and I F / > 2 , then i n p a r t i c u l a r 3 3 2 char F 3 . Hence x o ( x + l ) - x O X = x + x + ~ E L , s o x2+xEL. T a k e some f e n { O , l I a n d d e n o t e f - l * x 3 by p . Then po(x+f)-pox = x 2+fx+f2E L . 2 Since x +XEL a n d FcL, ( f - l ) x e L , s o by 7 . 7 9 ( b ) a n d t h e f a c t t h a t F i s a f i e l d we g e t xeL a n d a q a i n F[x] = L .

FIX].

87

224


7 . 8 7 R E M A R K B r e n n e r ( 1 ) h a s shown t h a t 7 . 8 6 ( b ) d o e s n o t h o l d f o r F = B 2 . See 7 . 9 8 ( b ) .

7 . 8 8 PROPOSITION

(Clay-Doi i s an

(a) (F,t)

(2)). a n d P ( F ) -group o f type 2.

F[x]-group

( b ) I f h ( 7 . 6 5 ) i s a n isomorphism ( c f . 7 . 6 6 ( b ) ) t h e n Fcx] i s 2 - p r i m i t i v e on ( F , t ) = F. (c) P(F)

i s a l w a y s 2 - p r i m i t i v e on F .

Proof. I f f , f ' E F , n a m e l y p o = ft:-'xo:

then 3 p o ~ F 0 [ x ] : p o ( f ) = f ' , The r e s t i s e q u a l l y o b v i o u s .

I f t h e r e a d e r i s s t i l l i n t e r e s t e d , he i s c o r d i a l l y i n v i t e d t o

a nearly complete trip t o the ideals of F i r s t we s e t t l e t h e q u e s t i o n , f o r w h i c h F be s i m p l e .

F[x] a n d P(F). F[x] happens t o

Fcx]

7.89 T H E O R E M ( S t r a u s ( 1 ) ) . Let F be i n f i n i t e . T h e n simple.

is

Proof. I f I 9 Fcx], I 9 {Ol, t a k e Some i c 1 , i 0. By 7 . 6 7 , i 6 , s o 3 f s F : i ( f ) = i o f 0. Hence i o f s I n F a n d 7 . 8 6 ( b ) i m p l i e s I = F[x].

+

+

S o t h e i n f i n i t e c a s e i s s e t t l e d a n d we t u r n t o f i n i t e f i e l d s . T o d o s o , we f i r s t d e t e r m i n e a l l f u l l i d e a l s ( 7 . 9 0 ) o f F cx ] a n d then we w i l l s e e ( 7 . 9 3 ) t h a t , i f c h a r F 2, a l l ideals o f FLx] are full ideals. 7.90 T H E O R E M (Menger ( 2 ) , Milgram ( l ) , Lausch-Nobauer

(l),

Straus (1)). Let F be a f i n i t e f i e l d and I a n i d e a l o f ( F [ x ] , t , * ) . Since (F[x],+,*) i s a P I D , I i s some p r i n c i p a l i d e a l ( p ) o f t h i s r i n g ( F [ x ] , + . . ) . Then t h e f o l l o w i n a c o n d i t i o n s are equivalent:

(a) I = (p) (b)

SEF[X]:

is a full ideal. p/po~.

225


( c ) There e x i s t

ml,.

and

kEN,

. .,mkf

Wo

~ ( x q

with

ml

Proof.

'k

,..., ( x

-XI

where

'2

nl

- ~ ) ~ ' ( )( xnq

I = ((xq

(then

l s n I < n2< . . . c n k

with

'1 p = 1.c.m.

...,

nl,

m2 -x)

IF1 = q

... n ( ( x q k'

)n

m - x ) ')).

(Straus). ( a ) -> ( b ) i s t r i v i a l . L e t C be the algebraic closure o f t h e

(c):

( b ) ->

f i e l d F.

-

p ( c ) = 0.

L e t c be i n C w i t h

m u l t i p l i c i t y of multiplicity

L e t m be t h e

t h e r o o t c . Then c i s a r o o t o f

>m

o f each

pos

p has a z e r o o f m u l t i p l i c i t y

(sEFLx1).

Hence

a t each element

2m

o f F(c) ( f i e l d extension o f F by adjunction o f c). A p p l y i n g t h e t h e o r y o f f i n i t e f i e l d s we s e e t h a t p : i s d i v i s i b l e by ( X ~ " - X ) ~ ,w h e r e n : = [F(c):F! dsF(c):

(x-d)"/p,

It

But

hence

TI

n

dcF(c)

(x-d)m

d i v i d e s p.

(x-d)m = (xq - x ) ~ .

dEF( c ) S t a r t i n g w i t h a r o o t c w i t h maximal

[F(c):F!

we

arrive successively a t (c). ( c ) =>

( a ) : By 7.77,

a r i g h t ideal.

Let s

i t s u f f i c e s t o show t h a t

R[x].

F o r m,n

induces the zero function.

But xq-x

E

degree zero f u n c t i o n i n P ( F ) , I f we d o t h i s f o r n = niy

E

IN

(p) i s ( x q n - X ) ~ sO

i s the lowestn whence ( x q - x ) l ( x q - x I m .

m = m .1 ( 1 2 i 2 k ) we g e t o u r

result. 7.91 REMARK The r e p r e s e n t a t i o n i n 7 . 9 0 ( c ) see Lausch-Nobauer

( l )Ch. ,

i s moreover unique:

111, 7 . 2 1 .

7 . 9 2 DEFINITION A p o l y n o m i a l p as i n 7.99 (Milgrarn (1)).

i s called saturated

87

226


7 . 9 3 T H E O R E M ( S t r a u s ( 1 ) ) . Let F b e a f i n i t e f i e l d . T h e n : E v e r y i d e a l o f F[x] i s a f u l l i d e a l c h a r F 4 2 .

P r o o f . ->: A s s u m e t h a t c h a r F = 2 . Isle show t h e e x i s t e n c e o f a n i d e a l I of FIX? which i s n o t a f u l l i d e a l : Let

n IF1 = : q

I:

Consider

I

4

and

F[x2]:

= {

1 a2ix2iIncINo,

i= O

a2i~F).

( ~ ~ t x ) ~ . F ~ x ~ ! t ( x ~ t x ) ~ . F [ x ] .

=

(F[x] , + , * )

since

Let p : = ( x q + x ) ' 1 2 (Fcx] , + , o ) :

(xqtx)2cI,

= x Z q + x 2 E F[x2].

but

~ . ( x ~ t x ) ~ $ I .

We s h o w t h a t

( a ) C l e a r l y , ( I , + ) i s a normal subgroup o f (F[x],+). i t suffices t o 2 c o n s i d e r u : = x o ( r + p t + p s ) - x n o r w i t h r , s E F[x] and t E F[x2]. I f n i s e v e n , u = ( r + p t + p 2 s ) ' - r n = z pntntp2nsn E I . I f n i s odd t h e n u = r " - l ( p t + p 2s ) + v w i t h v E ( p2 ) . So u = p ( t r n - ' ) + ( p r "Istv) E I , s i n c e ( b ) I n o r d e r t o show t h a t I SRF[x]

n

n-1

i s even.

( c ) I n p r o v i n g I a F [ x ] we h a v e t o s h o w t h a t f o r a l l r t E F[x 2 ] a n d a l l s , r E F [ x ] we g e t w : = ( p t + p 2 s ) o r E I . Now w = ( p o r ) ( t o r ) + ( p 2 0 r ) ( s o r ) . S i n c e c h a r F = 2 , t o r E F [ x 2 ] a n d p o r E F [ x 21. Now p ( p o r b y t h e s a m e a r g u m e n t a s i n the p r o o f ( c ) = > ( a ) of 7.90 a n d p 2 1 p 2 0 r as w e l l . Hence ( p 2 0 r ) ( s o r ) E p2F[x], and (por)(t.r) ( 3 :

Let

we n e t

E

pF[x2]F[x2j

c pF[x2].

This proves

W &

I.

I S F l x J . Ther, F o r a l l iEI and a l l p E =:x: i-p = -1( x 2 O ( i + p ) - x * o p - ( x 2 o ( i t 0 1 - x 2 0 0 ) E I . 2

7 . 9 4 R E M A R K T h e p r o o f o f " 7 . 9 3 2 than &(R[x]) = = (0:R).

zr

( c ) By 7 . 9 8 ( c ) , . d 2 ( Z 2 [ x ] ) c ( 0 : Z l 2 ) . ( d ) See a l s o 7.115 - 7.117. I n P i l z - S o ( 1 ) i t i s shown t h a t i f F? i s a f i e l d w i t h c h a r R f 2

21,2(R*[x])

then a l s o proved.

= {O}.

I n t h i s paper, the following r e s u l t i s

,$ 7 TRANSFORMATION NEAR-RINGS

238

7 . 1 2 8 T H E O R E M L e t R be a r i n g w i t h i d e n t i t y . Then P o ( R ) i s a r i n g i f f R i s a B o o l e a n r i n g . I n t h i s c a s e we g e t f o r all v

E

a,(

{0,1/2,1 y2}:

7

where i s t h e Jacobson r a d i c a l P ( R ) ) = ? ( P o ( R))+8.( R ) o f r i n g t h e o r y and Z ( R ) i s t h e i n t e r s e c t i o n o f a l l maximal submodules of t h e P o ( R )

- module R .

The s t a t e m e n t c o n c e r n i n g J v ( P ( R ) ) w i l l f o l l o w f r o m 9 . 7 7 . We rema i n a t P ( R a n d c i t e a r e s u l t o f ( K e l l e r - O l s o n ) . 7.129

= P =

I P ( Z n )I

$ -1 ) I

B(k)lP(Z

IP(ZD)l

=

In t h e r e , B ( k ) i s the smallest t E M with

pp.

b y r e p e a t e d appl i c a t i o n of 7.129.

There a r e numerous n e a r - r i n g s 7.130 _~

for k ? 2 and

between P ( A ) a n d M ( A ) .

DEFINITION L e t A be a n R - g r o u p a n d n E N .

(a) LnP(A):= (b) LP(A):=

if

fi

E

M(A)l

LnP(A).

'd

-

T s A , I T l < n , v d P E P(A):flT = P / ~ ? The e l e m e n t s i n L P ( A ) a r e c a l l e d

n EW

l o c a l polynomial f u n c t i o n s . Hence l o c a l p o l y n o m i a l f u n c t i o n s c a n be i n t e r p o l a t e d b y p o l y n o m i a l f u n c t i o n s , a n d we a r e b a c k t o t h e t o p i c s t r e a t e d o n p a g e s 1 3 3 / 1 3 4 and 2 1 9 / 2 2 0 . F i r s t we s t a t e 7 . 1 3 1 P R O P O S I T I O N L e t A be an R - g r o u p . Then L P ( A ) and e a c h L n P ( A ) are near-rings with P ( A ) < L P ( A ) . ... i L n P ( A ) i . 5 L ~ P ( A ) IL ~ P ( A )= C ( A ) Z L ~ P ( A ) = M ( A ) . Proof.

I t i s easy t o see t h a t L P ( A ) ,

L n P ( A ) and C ( A )

a r e subnear-rings of M ( A ) with P ( A ) s L P ( A ) s L n P cL,(A) = M ( A ) . Let n - 2 and a : b in A .

..


239

T h e n t h e r e i s some p c P ( A ) w i t h f ( a ) = p ( a ) a n d f(a) = p(b).

Hence f ( a ) : f ( b )

shown t h a t L n P ( A ) c C ( A ) . L2P(A)2C(A).

by 7.122

L e t f E C ( A ) a n d s u p p o s e t h a t a,b F A .

Let x zy iff x - y i

(b-a).

S i n c e a :b

we h a v e

f ( a ) z f ( b ) , whence f ( b ) - f ( a ) E ( b - a ) . i s some p EP,(A)

with p(b-a)

q:=

c P ( A ) f u l f i l l s :(a)

=

a n d we h a v e

I t r e m a i n s t o show t h a t

p9(=)+f(a)

=

By 7 . 1 2 3

f(b)-f(a).

f ( a ) and q ( b ) = p ( b - a ) + f ( a ) = f ( b ) .

there

Now

p(O)+f(a) =

=

Hence

f E L2P(A).

L o o k i n g b a c k t o 7.75 7.132

we ( r e - ) d e f i n e :

DEFINITION A i s c a l l e d ( a ) e o l y n o m i a l l y complete i f P(A)

=

M(A).

( b ) a f f i n e complete i f P(A) = C(A). ( c ) l o c a l l y p o l y n o m i a l l y complete i f LP(A) = M(A).

( d ) l o c a l l y a f f i n e complete i f LP(A) = C(A). Obviously, simple.

( l o c a l l y ) p o l y n o m i a l l y c o m p l e t e a l g e b r a s m u s t be

We r e m a r k t h a t o u r d e f i n i t i o n s d i f f e r - s l i g h t l y f r o m t h e

ones i n Lausch-NGbauer

( l ) , s i n c e we a r e o n l y c o n c e r n e d w i t h

p o l y n o m i a l f u n c t i o n s i n one v a r i a b l e .

In (II),

Nobauer c h a r a c t e r i z e d c o m p a t i b l e f u n c t i o n s on t h e

r i n g s Z a n d B n . We m e n t i o n w i t h o u t p r o o f . 7.133

THEOREM C ( Z ) = t f : Z + Z I f ( x ) C

7.134

~

ZE , A ( i ) = 1 . c . m .

1 ~ ~ A ( i ) ( ~ j w~h e) r ei , i= O

, . . . ,i .

EXAMPLE ( N o b a u e r ( 1 1 ) ) . f : Z + Z , b u t n o t a polynomial

7.135

o f 1,2

=

x

-t

1 4 2 -2( x + x ) i s c o m p a t i b l e ,

function.

EXAMPLES ( a ) A commutative r i n g R w i t h i d e n t i t y i s p o l y n o m i a l l y complete i f f R i s a f i n i t e f i e l d (7.75).

87

240


( b ) By L a g r a n g e ' s

theorem,

R i s locally polynomially

complete.

Z4, Z8 a r e p o l y n o m i a l l y c o m p l e t e ,

( c ) The r i n g s Z 2 ,

a f f i n e complete, n o t a f f i n e complete,

respectively

(see P i l z - so ( I ) ) . ( d ) By 7 . 1 0 4 ,

Z 2 and f i n i t e s i m p l e n o n - a b e l i a n

groups

are polynomially complete. ( e ) From 4.66

( a ) we k n o w t h a t L 3 P ( A )

LP(A) = M ( A ) ;

= M(A)

implies

hence A i s l o c a l l y p o l y n o m i a l l y complete

i n t h i s case. ( f ) A near-ring N i s polynomially complete i f f N i s f i n i t e and s i m p l e and i f N has e i t h e r n o n - a b e l i a n

addition

o r a b e l i a n a d d i t i o n w i t h a m u l t i p l i c a t i o n depending on b o t h arguments ( I s t i n g e r

-

Kaiser ( I ) ) .

We s h a l l i m p r o v e t h e s e r e s u l t s c o n s i d e r a b l y . d e f i n e a c o n c e p t due t o S.D.

F o r t h a t , we

Scott, which i s r e l a t e d t o

7.121

(see 7.140).

7.136

DEFINITION L e t N be a n e a r - r i n g and r an N-group

r

i s c a l l e d compatible i f f o r a l l y E T and n E N t h e r e

i s some m~

N w i t h n(y+li')-ny

N i s c a l l e d compatible

N-group

r

=

m6 f o r a l l 6 ~ " .

i f N has a f a i t h f u l

compatible

(we e x p r e s s t h i s b y s a y i n g t h a t N i s c o m p a t i b l e

on r). T h i s c o n d i t i o n means t h a t N a d m i t s a l l h o r i z o n t a l a n d " m a n y " v e r t i c a l t r a n s l a t i o n s (see P i l z ( 6 ) ) : P

T'

I

.'

/'

24 1


w i t h n(6) = n(y.6)

and m ( 6 )

( T h i s p i c t u r e shows a n r .

-_ 7.137

7.138

EXAMPLES M(i-1,

group

r),

(9.69;

r

N

=

E(&)-n(y) = n(y+6)-n(y). M(T)

\

~ ~ ( r ) , ~ ~ ~( r )~ ,

Mdiff(R)

(see 1.4

r.)

which i s compatible on

(a)),

~ ( f o ~r a ~t o p o(l o g ir c a l )

P(R) a n d M a ( r )

abelian) are compatible.

PROPOSITION An N - g r o u p

r

i s compatible i f f i t i s

c o m p a t i b l e as an N o - g r o u p .

r

If

Proof.

and y all

6

i s a c o m p a t i b l e N-group then f o r a l l n

No

E

~ t hr e r e i s some r n c N w i t h n ( y + S ) - n y = m 6 E

r.

D e c o m p o s i n g m i n t o m=mo+nic

1.13 g i v e s moR+mc&

=

6

E

r

0

and

according t o

mo6+mco o n t h e r i g h t s i d e .

C h o o s i n q 6 = o y i e l d s o = m o+mco. for all

for

Hence n ( y + 6 ) - n y

i s a c o m p a t i b l e No-group.

=

rno4

The c o n -

v e r s e i s even e a s i e r and o m i t t e d . 7.139

COROLLARY I f N i s c o m p a t i b l e o n on

7.140

r

t h e n No i s c o m p a t i b l e

r.

PROPOSITION E v e r y n e a r - r i n g N b e t w e e n P o ( A ) a n d C ( A ) ( i n p a r t i c u l a r , e a c h member o f t h e c h a i n i n 7 . 1 3 1 )

i s

c o m p a t i b l e on ( A , + ) . -___ Proof.

f

N a c t s on A i n t h e o b v i o u s and f a i t h f u l way. E

N and a

E

A.

T h e n 4 : = f o ( a-+ i d ) - f o a-,

f u n c t i o n which i s c o n s t a n t l y =a,

Let

where a i s the

i s i n N and

g ( b ) = f ( a + b ) - f ( a ) f o r a l l b E A. W i t h o u t p r o o f we m e n t i o n a r e s u l t o n c o m p a t i b l e N - q r o u p s . 7.141

12, b e c o m p a t i b l e on THEOREM ( L y o n s - S c o t t ( 1 ) ) L e t N I f N h a s t h e DCCL t h e n r i s " n i l p o t e n t b y f i n i t e " , Nr.

i.e.

r

has a n i l p o t e n t normal subgroup A such t h a t

is finite.

r/A

87

242


P r i m i t i v e compatible near-rings are studied i n Scott (17), i f t h e n e a r - r i n g NEX,

i t i s shown t h a t

Nr w i t h

t i b l e on

ACCL t h e n e i t h e r N

sparse i n a c e r t a i n topology

S.I).

Mo(r)

is finite or N i s

Nr).

( a r i s i n g from "zero sets" i n

Scott mentions i n p r i v a t e conmunications t h a t i n t h i s

r is

second case ( i f else

2

where

i s 2 - p r i m i t i v e a n d compa-

r

infinite)

i s divisible (cf.

We w i l l

r

e i t h e r has p r i m e exponent o r

9.19O(c)).

r e t u r n t o c o m p a t i b l e N - g r o u p s i n $9 9 ) .

N be

7 . 1 4 2 THEOREM - Let

R-group. Proof.

as i n 7.140

and A ( n o n - z e r o )

Let

B be a n o n - z e r o s u b g r o u p o f (A,+)

n o b c B f o r a l l n r N o a n d b~ B . o p e r a t i o n i n A and al Then ai

pi(b)

=

p . , q . E Po(") 1

, . . . ,a n

and bi

= qi(b)

by 7.123.

1

~

(

Let

&A,bl

iii

such t h a t

be an n - a r y

, . . . ,b n

f o r some b

EB. B and

E

L e t p be t h e zero-symmetric

p o l y n o m i a l f u n c t i o n z+(w(pl+ql

-

simple

T h e n N i s p r i m i t i v e on ( A , + ) .

,.. . , p n + q n )

-

,,... p p n ) ) ( z ) . T h e n w ( a , + b l ,... , a n + b n ) . . . ,a n ) = p ( b ) c B. H e n c e B i s a n i d e a l

- w(al,

o f A,

w h e n c e B = A.

S.D. S c o t t e n a b l e s u s t o d r a w i m p o r t a n t

A powerful r e s u l t o f

We w i l l m e n t i o n t h i s r e s u l t i n 9 . 1 7 0

conclusions. 7.143

THEOREM L e t A be a s i m p l e R - g r o u p .

dense i n M ( A )

P roof.

Then P ( A ) i s e i t h e r

(then A i s l o c a l l y polynomially complete) ( i n which case A i s a vector

o r P ( A ) i s dense i n M a f f ( A ) s p a c e o v e r Hom

(4).

Po ( A )

By 7 . 1 4 2

(A,A)

and F o ( A )

and 7.139,

compatible on(A,+).

i s a ring.

P o ( A ) i s p r i m i t i v e and By T h e o r e m 9 . 1 7 0 ,

Po(A) must

e i t h e r be a p r i m i t i v e r i n g o r dense i n Mo(A). 4.52

gives the result.

Now

I n order t o extend t h i s r e s u l t

t o some n o n - s i m p l e R - g r o u p s we n e e d m o r e i n f o r m a t i o n , 7.144

P R O P O S I T I O N L e t A be a s u b d i r e c t p r o d u c t o f R - g r o u p s

( i E I ) . Then f o r e v e r y f c C ( A ) t h e r e a r e u n i q u e Ai f i € C ( A i ) w i t h f ( . . . ,a i , . . . ) = ( . . . , f i ( a i ) , . . . ) for all

( . . . ,ai , . . . )

E

A.

If f

E

i E I.( F o r t h e p r o o f s e e e . g .

P(A) then fi P i l z (25)).

E

P(Ai)

for all


7.1_ 45 _ THEOREM _ _ _ ~L e t A b e a s

i n 7.144.

o f A i n P(A) then P(A)/Ji

=

243

I f Ji

denote the a n n i h i l a t o r

i s i s o m o r p h i c t o a sub-

P(Ai)

d i r e c t product o f the P(Ai)'s. P-r o o f .

We a s s i g n t o e v e r y f E P ( A )

fi E P(Ai)

i n 7.144.

f r o m P(A)

i n t o P(Ai)

with kernel

= z e r o map}

= Ji.@ i s

i f p i € P(Ai)

t h e n pi

) with aik) EA.

i d

(.

Ai

. . ,ai

(k)

k e r @= {p!pi

an endomorphism: i s a w o r d p . = w ( a i( 1 )

$

=6=

,. .. ,ai ( n ) ,

We r e p l a c e a i k ) b y some

,. . . ) = : d ! k ) A a n d fL\(xi ( 1 ) ,... , Z j n ) , i d A ) E E

1

Then p : =

the uniquely determined

T h i s g i v e s a homomorphism

i d

by idA. Ai P ( A ) a n d @ ( p ) = pi.

Now t h e h o m o m o r p h i s m t h e o r e m d o e s t h e r e s t o f t h e

job,

n Ji

together w i t h t h e remark t h a t

iEI T h i s shows t h a t e a c h P ( A i )

i s a homomorphic

=lo1 .

image o f P(A) i f

A i s a s u b d i r e c t product o f t h e Ails. The n e x t r e s u l t f o l l o w s 7.146

and 7.145.

COROLLARY L e t A b e a s u b d i r e c t p r o d u c t o f t h e s i m p l e Q-groups P(Ai)

Ai

NOW 7.124

( i

E

I).

Then P ( A ) i s s e m i s i m p l e a n d e a c h

i s dense i n M(Ai)

i n M - (Ai) atf

7.147

from 7.143

( i f Po(Ai)

( i f Po(Ai)

i s not a ring) or

i s a ring).

g i v e s us

THEOREM L e t A b e a s u b d i r e c t p r o d u c t o f s i m p l e i 2 - g r o u p s

Ai

( i

F;

I ) s u c h t h a t P o ( A ) h a s t h e DCCL.

Then P ( A ) i s t h e

d i r e c t sum o f f i n i t e l y many o f t h e P ( A i ) ' s .

7.148

e i t h e r equal

t o M(Ai)

( w i t h d i m Ai

finite).

( w i t h Ai

COROLLARY L e t A b e a s i n 7 . 1 4 7 Po(Ai)'s

are rings (c.f.

finite)

Each P(Ai)

i s

o r t o Maff(Ai)

such t h a t none o f t h e

7.128!).

Then P ( A ) i s f i n i t e

and 2-semisimple. Finally,

we c l o s e w i t h some r e m a r k a b l e e m b e d d i n g t h e o r e m s .

p r o o f s c a n b e f o u n d i n M e l d r u n i - P i l z - So ( 1 ) .

The

5 7 TRANSFORMATION NEAR-RINGS

244

7.149 -

T H E O REM ( a ) For every n e a r - r i n g N t h e r e i s a v a r i e t y v o f R-groups a n d some A

EV

with NGA

v-

[XI.

( b ) T h e r e e x i s t d . g . n . r ' s w h i c h c a n n o t b e embedded i n some r [ x ] (1' i n t h e v a r i e t y i o f g r o u p s ) .

s

( c ) E v e r y f i n i t e n e a r - r i n g c a n b e e m b e d d e d i n some P ( 1 ' ) for a finite,

simple non-abelian

group

r.

( d ) F o r e v e r y g r o u p T t h e r e i s some g r o u p A w i t h P ( A ) (A c a n b e c h o s e n a s ( T4[ x ]

r 4LX]G

,+)I.

(e) Not every abstract a f f i n e near-ring (see 9.71) b e e m b e d d e d i n some A a b e l i a n groups.

J(

[ X Iw,h e r e A

can

i s t h e v a r i e t y of

But every near-ring N i s abstract

a f f i n e i f f N i s i s o m o r p h i c t o some A

x

[XI, where

i s a v a r i e t y o f ( r i n g - ) modules. ( f ) W i t h a s i m i l a r i d e a as f o r ( a ) (namely b y a d d i n g u n a r y o p e r a t i o n s ) , one can f i n d f o r each c o m p a t i b l e nr.

N some R - g r o u p A w i t h M

and 7.137,

=

P(A).

Hence, b y 1.86

e v e r y n r . c a n be embedded i n a c o m p a t i b l e

one, e v e n i n a P ( A ) - t y p e one ( S . D .

Scott,

private

communication).

F o r many p u r p o s e s i t w o u l d b e v e r y v a l u a b l e t o h a v e a b e t t e r knowledae of t h e i d e a l o f a l l p o l y n o m i a l s which i n d u c e t h e zero function.

This i s j u s t the kernel o f the (near-ring)-epi-

rnorphism w h i c h a s s i g n s t o e a c h p o l y n o m i a l As we h a v e s e e n i n t h i s c h a p t e r ,

i t s polynomial function.

t h i s k e r n e l d e c i d e s i f one can

i d e n t i f y polynomials and polynomial f u n c t i o n s .

I t a l s o has se-

veral connections with the r a d i c a l s o f polynomial near-rings. I n Meldrum-Pilz

( 1 ) these questions are f u r t h e r investigated,

but they are f a r from being solved.

8 ) CONCLUDING R E M A R K S We c l o s e t h i s s e c t i o n w i t h some r e m a r k s c o n c e r n i n g q u e s t i o n s r e l a t e d t o p o l y n o m i a l a n d p o l y n o m i a l - 1 ike n e a r - r i n g s .

7d Polynomial near-rings 7.150

245

R E F~ I A R_ KS _

R,S& R[x] 2 S[x] i m p l i e s t h a t R 2: S ( t h i s f o l l o w s from 7 . 1 1 9 ) . He a l s o r e m a r k e d t h a t e a c h s u b n e a r - r i n g o f M ( F ) ( F a f i e l d ) which c o n t a i n s a l l c o n s t a n t f u n c t i o n s i s automatically simple. P ( R ) i s d i r e c t l y decomposable i f f this a p p l i e s t o R .

( a ) Nobauer ( 6 ) remarked t h a t f o r

( b ) If C i s a c o m p o s i t i o n r i n q a n d U i s a m a p C * C t h e n D i s c a l l e d a d e r i v a t i o n ( M e n g e r (3). MUller ( I ) , Lausch-Nobauer ( l ) , Nobauer ( 9 ) ) i f f o r a l l a,bcC: (1) D(atb) = D(a)tD(b)

("sum r u l e " )

( 2 ) D ( a . b ) = D(a).bta.D(b) ( 3 ) D(aob) = ( D ( a ) o b ) . D ( b )

("product r u l e " ) ("chain rule")

C l e a r l y t h e z e r o e n d o m o r p h i s m on C i s a ( t r i v i a l ) d e r i v a t i o n . R[x] has a l s o a n o n - t r i v i a l d e r i v a t i o n , namely the u s u a l o n e : D: p * p ' , All o n R[x] a r e v i v e n by D r : p rep', where r E R i s i d e m p o t e n t -+

.

(La.usch-Nobauer ( 1 )) N o b a u e r ( 6 ) showed t h a t t h e c o m p o s i t i o n r i n q M ( R ) h a s no d e r i v a t i o n s e x c e p t t h e t r i v i a l o n e . I f R i s a f i n i t e f i e l d , t h e same a p p l i e s t o P ( R ) (by 7 . 7 5 ) . I f R i s a n i n f i n i t e i n t e g r a l domain t h e n Y i i l l e r ( 1 ) showed e . 9 . t h a t i f ( R , t ) i s t o r s i o n - f r e e , t h e suma n d t h e c h a i n r u l e imply t h e p r o d u c t r u l e . ?!uller

stLdied a l s o " d e r i v a t i o n s " i n near-rings as well a s " i n t e r v a t i o n s " ( s e e ( 8 ) ) . C f . a l s o SeppXla ( 1 ) . ( c ) I n v e r t i b l e elements ( w . r . t . 0 ) are s t u d i e d i n LauschN o b a u e r ( 1 ) a n d Suvak ( ( l ) , ( 2 ) ) . T h o s e PER[X] s u c h t h a t 5;; i s b i j e c t i v e ( = i n v e r t i b l e ) a r e c a l l e d p e r m u t a t i o n p o l y n o m i a l s , were c o n s i d e r e d by many a u t h o r s a n d a r e p r e s e n t e d e x t e n s i v e l y i n Lausch-Nobauer ( 1 ) .

( d ) C l e a r l y R[x] and P ( R ) a r e i n g e n e r a l non-commutative n e a r - r i n q s . T h o s e p o l y n o m i a l s w h i c h commute w i t h a c e r t a i n f a m i l y o f o t h e r s w e r e s t u d i e d e . g . by Kautschi t s c h ( 1 ) and Lausch-Nobauer ( 1 ) . C a l l C c_ F[x! (F a f i e l d ) a P-chain ("permutable

37

246


chain") i f V c E C : jcI > O , v k E N j c E C : [c] = k a n d c , o c 2 = c 2 c c , f o r a l l c 1 , c 2 c C. E x a m p l e s : ( 1 ) The P - c h a i n o f p o w e r s { x , x 2 , x 3 , . . . I . -~

( 2 ) The P - c h a i n o f t e b y s h e v p o l y n o m i a l s I t l , t 2 , t 3 , . . I (where t n i s d e f i n e d via cos n$ = tnocos $ over F = Q a n d t h e n t r a n s f e r r e d t o F(x] for an a r b i t r a r y f i e l d F:

.

t.1 = x t 2 = - 1 + 2 x 2 -, t3 = -3xt4x 3 t q = 1 - 82~t 8 x 4 Also,

tnotm =

trim.)

I f 11 i s a l i n e a r p o l y n o m i a l and C i s a P - c h a i n t h e n C,: = ~ L o c ~ L - ~ l c t z Ci ls a P - c h a i n , t o o , c a l l e d a conjuqate P-chain. One c a n s e e ( t h e p r o o f s a r e n o t t o o e a s y - s e e L a u s c h N o b a u e r ( l ) , p . 156 - 1 5 9 ) : (a) I f

c i s a P-chain then C c o n t a i n s t o each e x a c t l y one c w i t h [c] = k .

kEIN

( 6 ) All P-chains o v e r a f i e l d F a r e conjucrates of

e i t h e r t h e P-chain o f powers o r o f t h e P-chain o f F e b y s h e v polytiomi a 1 s . ( e ) L a u s c h - N o b a u e r ( l ) , c h . 5 , c o n t a i n s more i n f o r m a t i o n on r[x] and P ( r ) For example, t h e c l a s s e s E k ( r ) o f a l l k - p l a c e f u n c t i o n s c r e n e r a t e d by a l l " k - p l a c e e n d o m o r p h i s m s on r " a r e c o n s i d e r e d ( " k dimensional composition qroups"). T h e s e a r e more e x a m p l e s o f d g n r . ' s , a n d r e s u l t s s i m i l a r t o o u r 6.33 and 7 . 4 6 a r e o b t a i n e d .

.

Fo[x] ( F a f i e l d ) . This i s a near-ring w i t h identity, b u t w i t h o u t divisors of zero. Fo[x] i s a l s o n o t r e c r u l a r $ 9 f ) ) . The

( f ) Heatherly ( 7 ) considered


ideals

Ik:

247

ktn = {

1

aix

InEIN

So

Fo[x]

i=k

descendinq chain.

aiEF1

0'

form a s t r i c t l y

does n o t f u l f i l l t h e

7.97).

DCC o n i d e a l s ( c f .

( 9 ) Nobauer ( 6 ) a l s o c o n s i d e r s t h e n e a r - r i n q s R(x) and R ( x ) o f a l l " r a t i o n a l p o l y n o m i a l s " and " r a t i o n a l

polynomial

functions".

7.75).

a f i n i t e f i e l d (cf.

R is it

decomposable i f f

( h ) T h e n e a r - r i n q s Ro[[x]] RE%

over

-

R(x)

i s directly

( c f . Remark ( a ) ) . of a l l formal

K a u t s c h i t s c h (1) - ( 8 ) and o t h e r s .

...,gn)

component o f

If

f E g:

II + II

. . ,X,,I~)~, defined

M : = ( ! I 0[cx,,.

i n t h i s s e t a composition

degree,

power s e r i e s

were considered by F r o h l i c h ( 9 ) , Cartan ( l ) ,

Frohlich ( 9 ) studied = fi(gl,

iff R i s

R(x) = M(R)

Again

(where

by

"0"

fi

denotes t h e i - t h

fEM.

all

c=>

fi

and

qi

t h e n one can c e f i n e i n

i n t h a t way t h a t

(M/:,t,o)

h a v e t h e same M/E

an a d d i t i o n

i s a near-ring

o f number-theoretic relevance. C a r t a n ' s r e s u l t was a l r e a d y m e n t i o n e d i n 1 . 1 2 . Graves-Malone

(3) looked a t the subnear-rina

N s a t i s f i e s t h e r i g h t Ore c o n d i t l o n ( 1 . 6 4 ) and i s integral.

( i )H e l l e r ( 1 ) d e f i n e d g e n e r a l i z e d p o l y n o m i a l s p i n

a

composition r i n g R by t h e p r o p e r t y t h a t f o r a l l f E R there are n €No =

c +clf+...+c,f 0

and c o n s t a n t cO, n

.

. . . ,c n €

R

C

with pof =

There e x i s t composition r i n g s i n which

every element i s a oeneralized polynomial, b u t n o t a polynomial. Anyhow,

t h i s s e c t i o n seems t o b e a w i d e f i e l d f o r f u r t h e r

research.

248

5 8 NEAR-FIELDS AND PLANAR NEAR-RINGS

T h i s c h a p t e r b r i n g s u p t w o i n p o r t a n t c l a s s e s o f n e a r - r i n n s . Isle s t a r t w i t h p e r h a p s t h e most i m p o r t a n t c l a s s , t h e n e a r - f i e l d s . A t h o r o u q h t r e a t m e n t w o u l d r e q u i r e n e a r l y a whole b o o k . B u t t h e r e a r e s e v e r a l e x c e l l e n t p r e s e n t a t i o n s o f p a r t s of t h i s t h e o r y ( e . 4 . Karzel ( I ) , Kerby ( 7 ) a n d I r ! a h l i n r l ( 6 ) ) s o t h a t we d a r e t o n i v e t h e t h e o r y D a r t l y b ! i t h o v t p r o o c s . r i r s t we c h a r a c t e r i z e t h o s e n r . ' s which happen t o be r i f . ' s . A f t e r showinq t h s t t h e a d d i t i v e a r o u p o f a n f . i s a b e l i a n w e rrive a s u p e r - s o n i c t r i p t h r o u c r h t h e r e l a t i o n s between n e a r - f i e l d s and geornet.1.y ( i n c i d e c e q r o u p s , c o o r d i n a t i s a t i o n o f p l a n e s , planar near-fields).

I n b ) we d e a l w i t h p a n a r n e a r - r i n q s . ; h e i r s t r u c t u r e i s 0) arc. d e f i n e d explored ( 8 . 9 0 , 8 . 9 6 , "blocks" aN+b ( a a n d i t i s shown t h a t a p l a n a r f i n i t e n e a r - r i n o t o n e t h e r w i t h i t s b l o c k s forms a t a c t i c a l c o n f i c u r a t i o n ( N , B ) . ? h e c a s e i s a balanced incomplete block desian i s when ( N & ) c h a r a c t e r i z e d i n 8.115 a n d s e v e r a l consequences a r e deduced. The a u t h o r t h a n k s Dr. G . Betsch f o r l e a v i n o h i n u n p u b l i s h e d lecture notes concernin? t h t s paragraph.

249

8a Near-fields

NEAR-FIELDS

al

1.) CONDITIO!4S T O B E A N-EAR-FIELD

We s t a r t w i t h ( c f . 1 . 1 5 )

8.1

PROPOSITION ---I f N i s a nf. then e i t h e r i s z e r o - s y mine t r i c

N

.

nC€Nc,

Proof. I f ---

+ 0,

nC

%

=

Mc(Z,)

-1 1 = nc n c = n C ,

then

Gr N

when c e

lEWc.

So nEN*: n = In = 1 , r e s t i s obvious.

hence

N = {0,1?.

The

8. 2--CONVENTION -

In a l l of o u r s u b s e q u e n t d i s c u s s i o n of nearMc(Z2) o f o r d e r 2 . f i e l d s we w i l l e x c l u d e t h i s s i l l y n f . ( c f . Malone ( 2 ) ) . Evidently, every near-field i s simple. We now c h a r a c t e r i z e t h o s e n e a r - r i n y s w h i c h a r e n e a r - f i e 1 d s : 8.3

THEOREM (Lioh

2 ) , Maxson ( l ) , B e i d l e m a n E q u i v a l e n t a r e f o r N&no:

1). F a i n ( 1 ) ) .

(a) N is a near-field. ( b ) lid $:

{o}

and

nEN*:

Nn = N .

( c ) N h a s a l e f t i d e n t i t y a n d NF1 i s Pi-simple. ( d ) N h a s a l e f t i d e n t i t y a n d N i s 2 - p r i m i t i v e on ( e ) N h a s a l e f t i d e n t i t y a n d N i s 1 - p r i m i t i v e on P r o o f . ( a ) -> ( b ) i s c l e a r .

(a): a,bcN* 3 a ' , b ' c N * : Thus a'(ab) = (a'a)b = b'b = a N i s integral. ( b ) ->

+

".

".

b'b = a A a'a = b'. 0, s o a b 0 and

250

$ 8 NEAR-FIELDS AND PLANAR NEAR-RINGS

T a k e some dEFI:. 3 eEN: ed = d . S o ( d e - d ) d = d e d - d d = 0 . From a b o v e , we n e t

de = d

N o w l e t n be Then

EN*. d ( e n - n ) = den-dn

b

Finally,

nEN*

This shows,that a near-field.

( c )

( a ) ->

8.4

3

n'EN":

(N",.) ( d )

4

R E -M- A R K ( L i g h ( 2 ) ) .

=

0,

whence

en = n .

n ' n = e. i s a oroup and

(e)

(N,t,-)

is

a r e obvious (observe 4.6)

Of c o u r s e , e . 0 .

( c ) i n 8.3 can b?

nEN" 3 n ' c N * : n ' n $. 0 r e p l a c e d by ( c ) ' : "Nd ==! { o } , a n d N N i s N - s i m p l e . " ( F o r ( c ) ' => ( b ) ==> ( c ) => ( d ) =o

=>

(c)'!)

W i t h o u t p r o o f we m e n t i o n t h e f o l l o w i n n r e s u l t s o f Liobi ( 2 ) a n d (1):

8.5

THEOREM Let N I 0 1 be a d g n r . . N i s a s k e w - f i e l d nEN" 33 n ' E N : n n ' n

8.6

t/

Nn

nEN":

=

=

n

N.

C O R O L L A R Y A f i n i t e i n t e q r a l d g n r . i s a commutative f i e l d .

+

A dgnr. N i s N-simple.

{Ol

with l e f t i d e n t i t y i s a f i e l d i f f i t

n $: 1 ,

8.7

T H E O R E M FIET)onn, i s a n f . e v e r y R E N , ( i n Eeidlenzn's sense - see 3 . 3 7 c ) ) .

8.8

R E M A R K See A n d r e ( 3 ) f o r a development of a t h e o r y o f

i s qr.

1 i n e a r a1 g e b r a o v e r n e a r - f i e 1 d s " a n d " n e a r - vec t o r - s p a c e s " ( c f . a l s o B e i d l e m a n ( 1 ) ) . S e e a l s o Grijger (l), P e l l e g r i n i ( 1 ) a n d Rado ( 1 ) a s well as 7 . 1 0 2 ( f ) . A very good survey on t h e a p p l i c a t i o n s o f n e a r - f i e l d s i s K a r z e l - K i s t ( 1 ) . "

8a Near-fields

251

2 . ) T H E A D D I T I V E G R O U P OF A N E A R - F I E L D

L e t t h e c h a r a c t e r i s t i c c h a r N of a n e a r - f i e l d N be d e f i n e d a s usual - (\{ahling ( 6 ) d e f i n e s char N : = c h a r Nd but this g i v e s t h e same ( s e e 8.23)).

-

Then one s e e s a s f o r f i e l d s :

8.9

P R O P O S I T I O N L e t N be a n f . a n d i n ( N , + ) . Then

o(1)

be t h e o r d e r o f 1

( a ) If

~ ( l )i s f i n i t e t h e n c h a r II = o ( 1 ) . ( b ) If o(1) i s i n f i n i t e then char N = 0 . ( c ) c h a r N i s e i t h e r 0 o r a prime.

For the foilowinq r e s u l t , c f . a n d apply 1 . 5 . 8 . 1 0 P R O P O S I.___ T I O N ( K a r z e l ( l ) , Maxson ( I ) , Linh-Neal Let N be a n f . . T h e n

(a)

tl

nc:N: ( n 2

(b)

v

n,n'EN: n ( - n ' )

=

(1)).

I n c { 1 , - 1 ] ) , (-n)n'

=

-nn'.

P r o o f . ( a ) : " n ' t n = 0 .

i s a qroup.

nEH: n o = 0.

v (f)v (e)

''*'I

i s a loop (with zero 0)

(N",.)

(d)

"+" and

N w i t h two b i n a r y

a set

n , n ' ,n"EN: n,n'~pI

3

(n+n')n" = nn"+n'n". d

n,n

,EN*

n"EN: n t ( n l + n ' l )

(n+n')+d,,nl

Near-domains

can be viewed as " a d d i t i v e l y n o n - a s s o c i a t i v e near-

fields"

8.75):

8.42

(cf.

R E M A R K A n e a r - d o m a i n w i t h a s s o c i a t i v e a d d i t i o n i s a nf..

I t i s n o t k n o w n i f t h e r e e x i s t n e a r - d o m a i n s w h i c h a r e no n e a r fields.

Anyhow,

t h o s e ones must be i n f i n i t e :

8 . 4 3 THEOREM A f i n i t e n e a r - d o m a i n

We d e f i n e f o r a near-dcmtiin N

i s a near-field.

T2(N)

a s i n 8.4'3 a n d g e t

n!'

$8 NEAR-FIELDS AND PLANAR NEAR-RINGS

260

8.44

THEOREM

( a ) For each near-domain N ,

T2(M)

i s sharply 2-transitive.

( b ) C o n v e r s e l y , f o r each s h a r p l y 2 - t r a n s i t i v e p e r m u t a t i o n g r o u p r on a s e t 1.7, M c a n be made i n t o a n e a r - d o m a i n s u c h t h a t r = T2(E4). 8.45 C O R O L L A R Y All f i n i t e s h a r p l y 2 - t r a n s i t i v e

groups a r e e x a c t l y t h e

T2(N)'s,

permutation where N i s a f i n i t e n f .

.

S o by 8 . 3 1 , 8 . 3 2 a n d 8 . 3 4 ,

a l l f i n i t e sharply 2-transitive permutation qroups a r e determined.

T h e r e e x i s t many c o n d i t i o n s u n d e r w h i c h a n e a r - d o m a i n i s f o r c e d t o b e a n e a r - f i e l d . They a r e e x c e l l e n t l y p r e s e n t e d i n Kerby ( 9 ) . We m e n t i o n o n l y o n e :

r i s a g r o u p t h e n I T : = { y c r l y 2 = 11 t h e s u b s e t o f t h e " i n v o l u t i o n s " o f r. Let ( 1 ~ 3 ' : = c Y ~ Y ~ I Y ~ , Y ~ E I ~ I .

8 . 4 6 NOTATION I f

denotes

8 . 4 7 T H E O R E V L e t r be a s h a r p l y 2 - t r a n s i t i v e p e r m u t a t i o n o r o u p on M a n d ( M , t , . ) " i t ' s " n e a r - d o m a i n ( 8 . 4 4 ( b ) ) . T h e n M i s a n e a r - f i e l d

(I,)'

5

r.

8 . 4 8 R E Y A R K S h a r p l y 3 - t r a n s i t i v e o r o u p s c a n be c h a r a c t e r i z e d

i n a s i m i l a r , b u t more c o m p l i c a t e d way b y a r o u p s o f t h i n a s l i k e " f r a c t i o n a l a f f i n e t r a n s f o r m a t i o n s " on c e r t a i n n e a r domains ( s o - c a l l e d " K a r z e l - T i t s - f i e l d s " ) . See Kerby ( 7 ) . See a l s o a l l S " - l a b e l e d i t e m s i n t h e b i b l i o s r a p h y .

6 . ) N O R M A L NEAR-FIELDS A N D I N C I D E N C E G P O U P S In o r d e r t o be a b l e t o f o r m u l a t e the c o n n e c t i o n s between n f . ' s a n d g e o m e t r y we d r i v e i n a n o t h e r c o u n t r y a n d r e c a l l some geometry. For a d e t a i l e d account s e e P,ndr@ ( 4 ) . C f . a l s o t h e a p p e n d i x t o Thornsen ( 1 ) .

26 1

8a Near-fields

8.49 D E F I N I T I O N L e t P b e a s e t a n d $ 5 2 ' . T h e p a i r (P&) i s c a l l e d a n i n c i d e n c e s t r u c t ure. (P,$) i s an i n c i d e n c e space provided t h a t

The elements o f

P a r e t h e n c a l l e d " p o i n t s " and t h o s e o f

L o f (a) i s c a l l e d t h e " l i n e determined by p , a " a n d d e n o t e d b y p9. If L , M & , s e t L / / M : ( L = M ) v Call (P$) deaenerated i f every s e t of 3 v (LnM = $). "lines".

p o i n t s i s o n a common l i n e . 8 . 5 0 D E F I i- Y I T I O t J Two i n c i d e n c e s p a c e s c a l l e d isomorphic if MsP:

h(M)Ef,'

P

or (if

= P'

c->

3 h:P

ME$.

and

+

and

(P,$) with

P'

h

b i j e c t i v e and

h i s t h e n c a l l e d an i s o m o r p h i s m

L = L')

an aut.omorphism.

8 . 5 1 DEFIPIITIOPI A s u b s e t S o f a n i n c i d e n c e s p a c e c a l l e d s u b s p a ce i f i t i s "convex",

s

=/=

t :

8.52 REMARK.

arc!

(PI,$')

-

if

i.e.

is

(P,t)

\I s , t ~ S ,

StES.

The subspaces o f an i n c i d e n c e space

an i n d u c t i v e Moore-system.

form

(P&)

Hence i t makes s e n s e t o s p e a k

about t h e "subspace aenerated by a subset of P".

8.53 DEFINITION A n o n - d e g e n e r a t e d i n c i d e n c e s p a c e

(P,$)

i s

c a l l e d an

(a) a f f i n e plane if (b) p r o j e c t i v e plane i f

: LoM

+ 8.

LE$

\

pcP

\I LEA:

3

ME&

ILI 1 3

EM and

A L//V.

L,M&:

Each a f f i n e p l a n e c a n be extended t o a p r o j e c t i v e p l a n e b y a d d i n g some p o i n t s .

Conversely,

one g e t s an a f f i n e

a p r o j e c t i v e one b y t a k i n g o u t one l i n e ,

p l a n e from


262

8 . 5 4 DEFINITION A s u b s p a c e o f an i n c i d e n c e s p a c e (P,;d) g e n e r a b l e by 3 p o i n t s ( n o t o n a c o m m n l i n e ) i s c a l l e d a plane i n ( P , , f ) . 8.55 DEFINITION An incidence space space i f each plane i n ( P L )

i s called a projective i s a ,projective plane. (P&)

be a p r o j e c t i v e s p a c e . B C P i s 8 . 5 6 DEFINITION L e t ( P , A ) c a l l e d a base o f (P,;L) i f B i s a minimal qeneratincr s e t for

(P,Z).

8.5 7THEOREM -

Each p r o j e c t i v e s p a c e h a s a ( n o n - e m p t y ) b a s e a n d a1 1 b a s e s a r e e q u i p o t e n t .

~

8 . 5 8 DEFINITION - I f B i s a base for the projective space P : = (P,$) t h e n d i m P : = ( B ( - 1 i s c a l l e d t h e --__ dimension o f P.

8.59 P ROPOSITION -

The a u t o m o r p h i s m s o f

p r o j e c t i v e space P ( t h e " c o l l i n e a t i o n s " ) f o r m a group Call ( P ) . d

8 . 6 0 DEFINITION A p r o j e L t i v e s p a c e (P,;C) i s called D e s a r g u e s i a n i f , whenever t w o " t r i a n g l e s " { a l , a 2 , a 3 ) and I b l y b 2 , b 3 1 ( a l , a 2 , a 3 , b , b , b E P ) a r e " p e r s p e c t i v e

w.r.t. a center

3

L1,L2,L3d

then

1

OEP"

( t h a t means t h a t

i e { 1 , 2 , 3 ) : O E L ~A a i E L i

~-

mnm, a l a 3 n m

some common l i n e L :

and

A

biELi)

m o m are in

8a Near-fields

263

\

bl /

L1

/

/

b;62/';

'

' X,'/ ' 3 4

I

I" /

43\

h

\

\ \

\'

2

pcM

M(I

If

M

with If

then

'

p-qEEa. L'

= n'+Bal

p-qEBa

nEaI.

L = L'. Since 0,lac8 a' i s an i n c i d e n c e space

implies that

L = n + B a c g and L.

qcL.

p,q~L'

and (N,$)

Now t a k e and

acN"

and

a l s o has t h e p r o p e r t y t h a t

(8.49). IN deqenerated.

( b ) and 8.11.

PEN.

If

Y:

(N,&) = piBa

i s not then

E h~ a s t i l e s a T e ~ r 3 ~ e r t +y e n


272

M ' = P+Bb M' = L

If

M = M'.

M'

If

Hence

+L 3

f o r some

M'n L = 0.

then

xcEa

3

beM".

p + g b = ntBa,

then

yeBb:

If

a = b, a

0,

p

-x+n =

a contradiction.

a = b

t1 = 1 4 ' .

and

(N,$)

8.94 REMARKS I t c a n b e shown t h a t t h e a f f i n e p l a n e 8.93(c)

r e s u l t can be o b t a i n e d i f t h e

Pa's

B a : = Ba ~ { o f u c o n n e c t i o n t o " @ ( I, I V ) - r i r o u p s " .

d e f i n e d as

i n

A similar

can be c o o r d i n a t i z e d b y a s k e w - f i e l d .

For a l l of

whence

PI = Ba+Bb.

b,

So

n - p = x+y.

= y + p c ( n + E a ) n (P+Bb) = Consequently again

so

are alternately

There i s a l s o a clos.

t h a t see Anshel-Clay

(2).

As C l a y p o i n t s o u t ,

t h e r e i s a l s o some r e l a t i o n t o " i n v e r s e p l a n e s "

(cf.

F e r r e r o ( 1 2 ) ) . I f N i s an i n t e g r a l o l a n a r n r . w i t h i d e n t i t y then N i s a s k e w - f i e l d o r isornornhic t o the n e a r - f i e l d {f : N

+

N l V m , n r N:

f(rnn)

rnf(n)}.

=

(2) o r Clay (10)).

8 . 9 5 E X A M P L ES (see Anshel-Clay

( a ) Every p l a n a r n f . w i t h more t h a n 2 elements i s a p l a n a r nr.. ( b ) L e t V be a normed v e c t o r space o v e r I R . =

11 w I I

v.

(V,+,*)

Then

Define

V ~ W :=

i s an i n t e n r a l p l a n a r non-

r i ncl. ( c ) L e t V b e a v e c t o r s p a c e o v e r IR the property that 4)(tv) Define

ta$(w). v*w: =

3

aeR*

II$(M) I 'Iav.

t j

and

tEIR,

Then

6:

V .,.

IR h a v e

t 2 o

(V,+,*)

VEV: i s a planar

n e a r - r i no. See A n s h e l - C l a y

o f t h e 6,'s ( d ) No

Vo(r)

(2)

f o r the oeonetric interpretations

as l i n e s ,

rays, hyperbolas

etc.

.

: i s d i s c r e t e , so (i-) i s a nf. with y(o) more t h a n 3 e l e m e n t s , w h i c h i s c e r t a i n l y n o t t h e c a s e . or M(r)

i s planar:

p l a n a r i t y would imply t h a t

So i n c o n t r a s t t o n e a r - f i e l d - t h e o r y , n o t p l a n a r i n aerieral

( c f . 3.77).

a finite

nr.

is

273

8b Planar near-rings

8 . 9 6 T H E O REM ( F e r r e r o (5), B e t s c h - C l a y

r

(a) Let

r.

automorphism aroup o f

N:

=

p

G

be a g r o u p and

I i d l

be a f i x e d - p o i n t - f r e e

If r i s f i n i t e t h e n e a c h

o f 1.4(b)

( r , + , a B )

inteqral iff

(1)).

i s a planar near-rinn.

N i s

i s t h e complete s e t o f a l l non-

IBilisIl

z e r o o r b i t s ( n o t a t i o n as i n 1 . 4 ) . l e t N be a p l a n a r n e a r - r i n n .

( b ) Conversely, aEN:

ga:

N

-+

n

N

+

na

point-free

(a):

Then

G:

flt

(ga\acN

=

%

C

=

1

$. { i d )

automorphism oroup

beti*,

F o r each

Proof.

.

Consider f o r i s a fixed-

o f (N,+).

Bb '

C o n s i d e r acidin t h e s i t u a t i o n o f 1 . 4 ( 5 ) .

IN/:( = [ G wiall23, s i n c e y E 6 a y = q 6 . So i t r e m a i n s t o s h o w t h e " p l a n a r p r o p e r t y " : Assume t h a t qy(E)

or

= g6(5)+q

(with

But

S o y

-qtid

y

= S*6trl,

or

T h i s means t h a t

6.

= n w i t h 9, A 9 6 $: i d ) : ( - q t i d ) ( C ) = - q i l ( r l ) .

(-q6+oy)(S)

-'q6

?: = i s bijective,

so t h i s e q u a t i o n has e x a c t l y

one s o l u t i o n : Suppose t h a t = -9(B)tB

9(a-a)

i s fixed-point-free Since (b):

r

= (-q+id)(B)

(-otid)(a)

and

and

q A id,

-n+id

i s finite,

-g(a)ta =

then

Since g

= a(a)-cl(B) = a-B. a = 6.

i s bijective.

If aspi*, C E N 3 X E N : p a ( x ) = xa = c So q a c A u t ( N , t ) and G = Cqa(acN*}

by is

8.88(b).

a group. C o n s i d e r t h e map

$:

(Bb,*) a-

Evidently,

If lbal

$(al)

-+

G

,

where

bcN*.

ga

UJ i s a h o m o m o r p h i s m .

= $(a2),

= lba2,

so

then al

= a*,

XEN:

xal

= xa2,

whence

and $ i s shown t o be a

monomorphism. .RI

Now t a k e s o ~ e g c , c ~ l l.

$(Ib c ) = albc

= 9,

Since

l b ccBb

by 8.9r)(c),

a n d t~ i s a n i s o m o r p h i s m .

274

$ 8 NEAR-FIELDS AND PLANAR NEAR-RINGS C;

i s fixed-point-free:

let

fulfill

qa(n) = n

f o r some n E N , n f 0, then 0 a n d n f u l f i l l x a = x * l a + O ( 8 . 9 0 ( e ) ) . S o a 5 1a ' which means that XCW: q a ( x ) = xa = x l a = x , from which we

v

deduce t h a t

ga = i d .

8 . 9 7 R E M A R K ( B e t s c h - C l a y ( 1 ) ) . T h i s shows t h a t ( s i m i l a r t o t h e I

s i t u a t i o n in planar n e a r - f i e l d s ) every f i n i t e planar nezrr i n q c a n be c h a r a c t e r i z e d by some p a i r ( ? , G ) o f q r o u p s , where G .C. { i d 1 5 A u t r i s f i x e d - p o i n t - f r e e . S o e v e r y f i n i t e p l a n a r n e a r - r i n a determines a Frobenius oroup (8.79) a n d conversely ( c f . also Ferrero ( 5 ) ) , a n d the c o n s t r u c t i o n o f a p i a n a r r i e c r - r i n o CI: a n i v e n a d d i t i v e

g r o u p 'I i s n o t h i n q e l s e t h a n t h e c o n s t r u c t i o n o f a nont r i v i a l f i x e d - p o i n t - f r e e automorphism n r o u p o n r . Cf. 8 . 1 2 4 , H e a t h e r l v - n l i v i e r ( 3 ) a n d Adler ( 1 ) .

-8 . 3 8 C O R O L L A R Y ( B e t s c h - C l a y ( 1 ) ) L e t Id he a f i n i t e p l a n a r n e a r _ I

r i n g a n d l e t G be a s i n 8 . 9 6 . T h e n

( a ) IGj d i v i d e s (b) (N,t)

Proof.

I F 1 1 - 1.

i s nilpotent,but not necessarily abelian.

( a ) i s c l e a r from 8 . 9 6 ( b ) a n d 8 . 9 0 ( b ) ,

( b ) follows from ( T h o m p s o n )

and ( c f . 0 . 3 3 ( b ) -> ( a ) ) .

See a l s o 8 . 1 2 4 . The l a s t r e s u l t i s i n some o t h e r way r e m a r k d b l e :

planar nearr i n g s a r e " n o t f a r away from b e i n n n e a r - f i e l d s " ( c f . 8 . 8 8 l b ) ) . B u t t h e y a r e f a r enouqh t o have n o n - a b e i i a n members i n c o n t r a s t t o 8 . 1 1 . We need

8 . 9 9 DEFINITI9N ( F e r r e r o (5), S z e t o ( 3 ) ) . A nr. N i s c a l l e d s t r o n q l y u n i f o r m i f ij n E N : ( 0 : n ) = ( 0 1 o r ( 0 : n ) = M , but

3

mEN:

(0:m)

=

{O).

For t h e f o l l o w i n g r e s u l t , c f . a n d Olivier ( 3 ) .

Ferrero ( 5 ) ,

Heatherly-Olivier ( 3 )


8.100

THEOREM ( F e r r e r o ( 5 ) ,

Clay (ll),

275

S z e t o (3)).

( a ) L e t N be a p l a n a r n r . . Then N i s s t r o n g l y u n i f o r m ,

the multiplication i s not t r i v i a l !1.4(b)) a n d a l l non-zero o r b i t s o f G ( s e e 8 . 9 6 ( b ) ) a r e p r i n c i p a l ( t h a t means t h a t f o r a l l x , y o r b i t t h e r e i s e x a c t l y one

i n t h e same n o n - z e r o r j ~ G with o(x) = y).

( b ) C o n v e r s e l y , i f N i s a f i n i t e nr. which i s s t r o n q l y uniform, has n o n - t r i v i a l m u l t i p l i c a t i o n a n d the p r o p e r t y t h a t e v e r y n o n - z e r o o r b i t u n d e r C; ( d e f i n e d a s i n 8 . 9 6 ( b ) ) i s p r i n c i p a l , t h e n ii i s p l a n a r .

a€/\, qa = 6 a n d ( 0 : a ) = N . I f aEN", and (0:a) = { O l , hence N i s strorinly 1N/:i23, t h e m u l t i p l i c a t i o n cannot uniform. Since be t r i v i a l . G i s fixed-point-free (8.96), so a l l o r b i t s are principal (cf. 4.28).

Proof.

_I.-I

( a ) If

gaEAut

(b)

9,:

FI

S i n c e PI i s f i n i t e a n d s t r o n r l l y u n i f o r m , a l l x + xa a r e e i t h e r = 6 o r a u t o m o r p h i s r n s .

(Observe t h a t

Ker q a = ( 0 : a ) ) .

L e t G be t h e orolJp

Since a l l o r b i t s are p r i n c i p a l , G i s f i x e d - p o i n t - f r e e . Since

o f a l l t h o s e automorphisms.

trivial, R E tlARK -8 . 1-0 -1 ~

G

{id}.

$.{O!

i s not

Now a p p l y 8 . 9 6 ( a ) .

( S z e t o ( 3 ) ) . 8 . l O O ( b ) does n o t hold i n the i n f i n i t e

( Z , + ) @ (??,+) and d e f i n e (n,m)c(n',m'): T h e n TI: = ( T x Z , + , * ) i s a n i n f i n i t e s t r o n o l y uniform P r . . * i s not t r i v i a l a n d a l l non-zern o r b i t s + r e principal. On t h e o t h e r h a n d , N i s n o t p l a n a r , f o r (2,O) (0,O), but x(2,O) = x(O,O)+(l.l) has no s o l u t i o n . C f . a l s o r,. Betsch's report i n the " Z e n t r a l b l a t t f u r Mathematik".

c a s e : Take

2

= nl(n,m).

I n F e r r e r o - C o t t i - P e l l e g r i n i ( 1 ) i t i s shown t h a t 2 i f N i s p l a n a r t h e n U =N. F o r I : F i r s t l e t a N t b = a ' N t b ' . If b = b' then aN = a'FI a n d we a r e i n c a s e ( a ) . So s u p p o s e t h a t b b'. F r o m a11 = a ' r i t ( b ' - b ! we g e t s o m e n E N w i t h 0 = a ' n t ( b ' - b ) . b b' i m p l i e s t h a t nEN*. So a ' n = b - b ' . Sinilarly, 3 n'EN'+: a n ' = b ' - b . Hence 0 b - b ' E ( a ' I i * ) n (-ail*), whence a'N* = iw = -aN by 8.105. So a N t b = - a N t b ' . Consequently 1 n"EN*: b ' = a n " + b , s o a N t b = - a N t b ' = - a N t a n " + b , w h e n c e aN = - a N t a n " = - a N t q n , , ( a ) . A p p l y i n q a n- ,1. g i v e s aN = - a N + a , s o aEC1(N). By s y m m e t r y , a'EC1(N) and ( b ) i s shown.

+

+

aN+b = a ' N t b ' . So a s s u m e ( b ) . L e t x E a l N t b ' . We h a v e t o show t h a t tlt xEaNtb. I f x = b'EaK t b , x ~ a N t b . I f x f b ' , 3 n,n',n":Y*: x -- a ' n s b l = - a n ' t ~ ' = - a n ' t a n ' ' + ~ . S i n c e aN I ( N ,+) , b y 8 . 1 0 8 , - a n ' + a n " E a N , whence xca'l-b. Ti.e c n n v e - q c i n c ' l ~ 5 3 7 1 - 5 shol.pn s i ~ i l a r l y . l a n d and r r k .

ksv-1

then

btv

( "Fisher's inequality") A -

BIBD's a r e an e s s e n t i a l t o o l i n e x p e r i m e n t a l d e s i o n s . The f o l l o w i n g example s h a l l i l l u s t r a t e t h i s a n d p r o v i d e enourlh motivation f o r t h e r e a d e r t o endure a l s o t h e next paoes.

8.115

A P P L I C A T I O N Suppose you have b k i n d s of f e r t i l i z e r s and w a n t t o t e s t some c o m b i n a t i o n s o f r f e r t i l i z e r s always o n t h e same number k o f e x p e r i m e n t a l f i e l d s . Take some B I B D with parameters (v,h,r,k,X), a n d d i v i d e t h e whole e x p e r i m e n t a l a r e a i n t o v p a r t s .

(P,B)

S i n c e IBi b = number o f f e r t i l i z e r s , c a n be w r i t t e n a s 3 = C B 1 , B 2 , . . . , B b l . Give t h e f e r t i l i z e r number i o n e v e r y f i e l d o f t h e b l o c k B i . Then: ( a ) every f i e l d c o n t a i n s e x a c t l y r d i f f e r e n t f e r t i l i z e r s , ( b ) every f e r t i l i z e r i s applied on e x a c t l y k d i f f e r e n t f i e l d s , and ( c ) e v e r y p a i r of d i f f e r e n t f i e l d s has e x a c t l y h k i n d s o f f e r t i l i z e r s i n common. i t i s a non-trivial 8 . 1 1 6 R E Y A R K S Of c o u r s e , g i v e n b , r , k , problem how t o q e t a BIBD w i t h s u i t a b l e p a r a m e t e r s . I n g e n e r a l , i t i s a n open q u e s t i o n w h e t h e r f o r e v e r y quintuple ( v , b , r , k , h ) o f n a t u r a l numbers which f u l f i l l the c o n d i t i o n s o f 9.!14 t h e r e E X - s t s a BIEO with t h e s e p a r a m e t e r s . We w i l l n o w a p p l y p l a n a r n e a r - r i n o s t o q e t nc?w c l a s s e s o f E 1 I : O ' s .

281


The e f f i c i e n c y o f a B I B D c a n be i n t e r p r e t e d e c o n o m i c a l l y i n t h e example above. B I B D ' s o f e f f i c i e n c y 1 0 , 8 5 a r e u s u a l l y c o n s i d e r e d t o be " g o o d " . Vany o f them a r e l i s t e d i n (Cochran-Cox) Balanced complete block designs a r e u s u a l l y " r a t h e r i n e f f i c i e n t " . T h i s i s the reason f o r lookinq a t t h e incomplete ones.

.

8 . 1 1 7 THEOREll ( F e r r e r o ( 1 2 ) ) . L e t N b e p l a n a r w i t h I N [ = : V E N , Denote b y 3 t h e s e t o f a l l b l o c k s ( 8 . 1 0 4 ) . L e t a l (a2) be t h e number o f n o n - z e r o c r b i t s o f (N,t) under G c o n s i s t i n o of elements of C 1 ( N ) (not o f C1(N), r e s p e c t i v e l y ) ( c f . 8 . 1 1 0 ) . Then (11,B) i s d t a c t i c a l configuration w i t h parameters

P-r o o f . The f i r s t p a r a m e t e r i s c l e a r . c o m p u t e t h e number o f d i f f e r e n t b l o c k s a n d a p p l y 8.111: The number o f b l o c k s a N t b w i t h a c C l ( N ) i s

We

y -, lGOl

t h e one of t h o s e w i t h

atCl(N)

(case ( b ) ) i s

u2-v.

Now a p p l y 8 . 1 0 6 t o net k = \ G o / i n 8 . 1 1 2 ( d ) . N e x t o b s e r v e t h a t t h e number r n o f b l o c k s c o n t a i n i n r r an e l e m e n t n E N i s t h e same f o r e a c h ncN, s i n c e i t e q u a l s t h e number o f b l o c k s c o n t a i n i n o 0. Now we know t h a t ( N , B ) i s t a c t i c a l a n d we c a n a p p l y 8 . 1 1 4 ( a )

c o u r s e , t h i s c o u l d be a c c o m p l i s h e d d i r e c t l y , t o o ) . Observe t h a t v = ( a l + a 2 ) [ C 1 t 1 . N o t h i n q i s more n a t u r a l now t h a n t o a s k , u n d e r w h i c h c o n d i t i o n s (ti,%) i s a 3!93. T + e T e f c t ) i e o r o i a n s w e r s t i i i : q u e s t i o n , t t l u s b r i n q i n a j o y and happin,ess i n t o o u r l i f e .

282

5 8 NEAR-FIELDS AND PLANAR NEAR-RINGS

8.118 THE9REI.I ( F e r r e r o ( 9 ) - ( 1 2 ) ) . Let ( N , ? 3 ) be a s a b o v e . (N,B) i s a B I B D C 1 ( N ) i.1 ( t h e n h = 1 ) o r (then X = l G o l ) . C1(N) = {Ol

P r o o f . - > : I t d o e s n o t seem t o be p o s s i b l e t o d e d u c e r Ck - 1 ) E Z ( 8 . 1 1 4 ( b ) ) . t h i s f r o m t h e f a c t t h a t -vz-So we have t o work. C a l l ( f o r a , b E F I ) a , b e q u i v a l e n t i f aH = b l I ( a a n d b a r e t h e n i n t h e same o r b i t i l n d e r G ) a n d d e n o t e t h i s by a x h . Ide n e e d a lemma. N be p l a n a r a n d n ' , n " be E N , n ' ?= n " . L e t A : = A~ be t h e number o f b l o c k s B w i t h n',n"cR ( " b l o c k s thr-ouoh n ' a n d n " L e t 11 b e t h e number o f d i f f e r e n t r e p r e s e n t a t i o n s of n: = n'-n" as 3 d i f f e r e n c e of two eqcriva1ent elements n o t contained in C l ( N ) . Then: If nEC1(N) then h = p + l . I f ntC1(FI) t h e n A = 1 1 + 2 .

-Lemma: L e t

'I).

P r o o f o f t h e Lemma: F i r s t o b s e r v e t h a t i f t h e block aNtb contains D a n d n ( = n ' - n " ) then

aI.I+(b+n")

contains n ' and n".

X i s t h e number o f b l o c k s t h r o u q h

r)

hence and

n (+ 0). {O,nir'aN+b be s u c h a b l o c k .

How many d i f f e r e n t b o c k s w i t h

e x i s t ? Let

aN+b

whence nii = C a s e ( 1 ) : I f b = 0 , nEaH*, ) a n d t h ere i s only = aN by 8 . 1 0 9 ( one p o s s i b i l i t y t o have I 9 , n l E a ' R f o r some a'Er4'. Case ( 2 ) : b = n . Then OEali+n, n E -aiJ, whence aN = -nN. So t h e r e i s a a a i n j u s t one block t h r o u o h 0 and n .

*

n. 3 nlEN : 0 = anl+b. C a s e ( 3 ) : 0 -f b Hence a ' i = - b r l , a n d d H + h = - b N + b . So i f n i s a d i f f e r e n c e a s s t a t e d i n t h e l e m m a , t h e blockc, ? n t o r i s i ? e r ? ' l ~ ?

283


have t h e form -cN+c. Conversely, f o r the block -bN+b +It we a e t , s i n c e n E a N t b , 3 n 2 E i l : n = = -bn2+b, which i s a r e p r e s e n t a t i o n o f n a s a d i f f e r e n c e o f two e q u i v a l e n t e l e m e n t s of bN. If bEC1(N), bFls(i4,t) i m p l i e s nEb1.I. S i n c e a l s o O c b N , we a r e i n c a s e ( I ) , a c o n t r a d i c t i o n t o 0 $. b $. n . S o l e t b be & C 1 ( N ) . Then - b l . I t b i s n e i t h e r in case (1) nor ( 2 ) n o r equal t o sortie o t i e r - b ' X : t i ' containinn 0 2nd n, b u t w i t h b 4 b ' by 8 . 1 1 1 . S o i n c a s e ( 3 ) a r e j u s t a s many b l o c k s n o t i n ( 1 ) and ( 2 ) a s t h e r e a r e r e p r e s e n t a t i o n s of n of the described kind, namely u . So t h e r e s u l t f o l l o w s i f one o b s e r v e s t h a t the two blocks i n (1) a n d ( 2 ) coincide i f f nEC1(N).

P r o o f o f t h e t h e o r em. 8y t h e planar n r o p e r t y , +t nEN* n',n"tFi , n o n" 3 XE~I: n = xnl-xn". So n has ~ G l - ( l ~ \ - l s)u c h r e p r e s e n t a t i o n s (when varyirtq n ' , n " ) . Now t a k e some a r b i t r a r y q E G . -1 Then n = x n ' - x n " = q n l ( ~ ) - q n , , ( x )= ( a n r o a ) ( r l ( x ) ) -(qn.oa)(q-l(x)), p r o v i d i n c r a l l o t h e r ways t o w r i t e n a s a d i f f e r e n c e of e q u i v a l e n t elements. So t h e r e

+

are j

lrst

' G " ( l G ! 1GI - q-

1

d i f f e r e n t ways t o

IGI w r i t e n as s u c h a d i f f e r e n c e .

( a ) If n&C1(Y) a n d n = a-b (azb) then a , b a r e b o t h t$Cl(Fi). For i f e.o. aEC1(N) then bEC1(N), whence a - b c C 1 ( N ) by 8 . 1 0 8 a n d 8 . l I n . By o u r l e ~ r , ; , h = : , t 2 = ( I G ' - l j t 2 = , G ' t I .

284


(b)

If

as

n = a-b

ncC1(N)

\GI-1

then the

with

a%b

ways t o w r i t e n

n = a-(a-n)

are exactly

w i t h aEnW*\H\In). F o r nN i s ( 8 . 1 0 8 ) a n a b e l i a n group of order l G o l ; s o V acnN*\(\(nl; a - n = = - n t a Q a. Observe t h a t a and (a-n) a r e i n C1(N). So none o f t h e ! G I - 1 differences o f equivalent e l e m e n t s g i v i n g n a r e a s d e s c r i b e d i n t h e lemma, p = 0,

whence

and

X = 1.

I t may h a p p e n t h a t ( i n 8 . 1 1 7 a n d 8 . 1 1 8 )

nor

Cl(rl)

{Ol

=

neither

C1(N)

= N

(see Betsch-Clay ( I ) ) .

One c a n e v e n s a y m o r e ( s e e F e r r e r o ( 1 2 ) ) : 8.119

(N,p)

REMARK L e t

Cl(N)

If

be t h e BIBD o f 8.117/8.118.

then

= N

(N,t)

i s elementary a b e l i a n (8.108)

a n d t h e r e i s some f i n i t e f i e l d F s u c h t h a t

= (F2,X) space,

o f 8.74;

(N,B)

(N,$)

=

can be considered as affine

and t h e b l o c k s a r e j u s t t h e l i n e s o f t h i s space.

L o o k i n g a t t h e o t h e r c a s e ( w h i c h b r i n c l s u p p o s s i b l y new d e s i c r n s )

yields f i r s t 8 . 1 2 0 COROLLARY ( F e r r e r o ( 1 2 ) ) . Let

pa/[NI. = lGol.

Then

pa, w h e r e pciP a n d o f 8.117 i s a BIBD w i t h k = X =

(N,3)

P r o o f . Assume t h a t

3

i s PEP, ( 8 . 1 0 8 ) , a c o n t r a d i c t i o n . Hence and 8.118 g i v e s t h e r e s u l t . nEN*:

elementary abelian, and

pa/lN!

CI(N) See F e r r e r o ( 1 2 ) , planes.

Cf.

L e t N be a f i n i t e p l a n a r n r .

\ G o \ have n o t t h e f o r m

=

{Ol

ncC1(N).

so

l n N l = pa

Then

nN

with

Teorema 8 f o r t h e c o n n e c t i o n t o f i n i t e f48bius

a l s o Anshel-Clay

A n o t h e r way t o r e a c h t h e c a s e

(1).

C1(N)

=

!Ol

i s the followino.


8 . 1 2 1 COROLLARY ( c f .

285

Ferrero ( 8 ) ) . L e t N be a f i n i t e i n t e g r a l

planar nr. without subnear-fields.

T h e n t h e same c o n c l u -

s i o n as i n 8.120 h o l d s . P r o o f . Suppose t h a t

ncN*

i s i n

an a b e l i a n subaroup o f (nN)"z

C1(N)

E

beOn c a n

n:I

a contradiction.

{Ol

=

Bn:

b y 8.90 ( a ) . i s a a r o u p and nll i s a

n

((ntl)",.)

s u b n e a r - f i e l d o f Id,

i s

(nN)*

while every

b = 1 b = nn-lb

Consequently,

nN

Then

b y 8.108.

(N,t)

B n i s c l c a r f r o F 8.89,

be w r i t t e n as

tience

C1(N).

and t h e r e s u l t f o l l o w s f r o m

8.118. I n (ti),

Ferrercj constructs

( I N / ,6) IN1 = v ) .

BIBD's

f r o m n e a r - r i n g s 14 w i t h v v 1) ( v , --&., 3, 3) (where

having parameters

1,

9,

B o t h c a s e s i n 8.118 c a n b e o b t a i n e d by t h e f o l l o w i n q n e a r - r i n o s : 8 . 1 2 2 COROLLARY ( C l a y ( 1 1 ) ) .

Consider t h e planar nr.

(F,+,*t)

o f 8.103. (a)

m t = p -1

If

8.117 (P (b)

n 9

-

msn

f o r some

pn(pn-l) --m p (p"-1)

pn-1

P

1

9 -

pm-1 n

(p

Proof. F i r s t observe t h a t t + l = 1G

0

aeF*.

= pm-1

Then

pn(pn-l) t

rn

= x,

(msn). (F,+):

follcws

is a

(t+l)(pn-l) t

9

1

( o f 8.103),

Set has

B

E:

=

B uf'l}.

t + l = pm

from 8 . 1 ? 3 ,

elements

consists o f a l l

hence b e i n q a subrrroup o f

T h i s i s easily t r a n s i e r r e d t o (b)

(F,B)

t = / G I = i6l

a*F = a B

and i s a subaroup o f

xp

,

then

1.

( a ) L e t t be

wit4

as i n

.pm4

BIBD w i t h p a r a m e t e r s ttl, t + l ) .

Take

-

(F,B)

m , 1).

I f t i s n o t o f t h e form

so

then

i s a BIBD w i t h parameters

aE.

XCF

(F,+).

Now a p p l y 8.!!8.

286

$ 8 NEAR-FIELDS AND PLANAR NEAR-RINGS

8.123 REMARK Observe t h a t one can efficiency

E =

pn-pn--"

qet

RIBD's o u t o f 8.122 w i t h

(in (a) )

and

E =

pn- 1

p n . t_(pn-1) ( t+l)

( i n ( b ) ) , which i s c l o s e t o 1 f o r l a r g e n . 8.124 R E M A R K B I B O ' s c a n a l s o be c o n s t r u c t e d f r o m n o n - a b e l i a n f i n i t e p l a n a r n e a r - r i n q s ( s e e Clay ( 1 1 ) ) . D e f i n e on B 7 ~ H 7 x i Z 7 an a d d i t i o n "8" by

(a,b,c) 8 (d',b',c'):

=

(a+a', b t b ' , c+c'+a'*b).

N be d e f i n e d v i a q ( a , h , c ) : = ( 2 a , 2 b , 4 c ) . L e t q:N T h e n ( B . H . rleuriann ( 2 ) ) (N,$) i s a non-atelian n ! - ~ u p 2 and G : = ! i d , q , o 1 i s a f i x e d - p o i n t - f r e e automorprlisri group of ( N , @ ) . Clay goes 8 . 9 6 ( a ) q i v e s some p l a n a r n e a r - r i n n ( N , G I , s B ) . on t o p r o v e t h a t i s a B I R D , o f c o u r s e v!ith k = X a n d C 1 ( N ) = {Ol ( t h i s f o l l o w s f r o m 8 . 1 2 0 ) . Clay a l s o q e n e r a l i z e s t h i s example. -+

(N,a)

S e e B e t s c h - C l a y ( 1 ) f o r a n e x c e l l e n t summary o f t h e t h e o r y o f p l a n a r n e a r - r i nqs t o c l e t h e r wi t h new r e s u l t s ( e . 0 . c o n n e c t i o n 5 t o p a r t i a l l y balanced incomplete block desiclns) a n d h i n t s f o r f u r t h e r r e s e a r c h . See a l s o C l a y ( 1 7 ) , ( 1 3 ) .

287

3 9 MORE CLASSES OF NEAR-RINGS

a ) c o n t a i n s commutativity theorems s i m i l a r t o the "n(x)-theorem" o f J a c o b s o n and t h e " n ( x , y ) - t h e o r e m " o f H e r s t e i n i n r i n q t h e c r y . Our d i s c u s s i o n i s d o n e i n t h e w o r l d o f I F P - n e a r - r i n a s ( t h a t a r e n r . ' s N w h e r e a b = 0 i m p l i e s a n b = 0 f o r a l l nc:i). A d q n r . with the "n(x,y)-property" i s a convutative r i n o . p - n e a r - r i n q s a n d Boolean n r . 3 a r e a l s o c o n s i d e r e d ( a s s p e c i a l cases). N e x t , we s t u d y n r . ' s w i t h o u t n i l p o t e n t e l e m e n t s . They a r e ( i f in s u b d i r e c t p r o d u c t s o f i n t e n r a l n r . ' s which a r e s t u d i e d i n p a r t 2 ) o f b ) . The f i n i t e i n t e a r a l n e a r - r i n a s a r e planar i f f they are n o t " t r i v i a l " . Special inteqral n r . ' s ire c a l l e d " n e a r - i n t e g r a l domains". T h e i r c h a r a c t e r i s t i c i s zero o r a prime. c ) c o n t a i n s a d i s c u s s i o n of a f f i n e n r . ' s ( i . e . a G e n e r a l i z a t i o n o f n r . I s o f t y p e M a f f ( V ) ) . We e x a m i n e t h e i d e a l s t r u c t u r e , the radicals and n r . ' s constructed o u t of affine n r . ' s . F u n d a m e n t a l f o r t h e s e n r . ' s i s t h e f a c t t h a t ri0 i s a r i p 9 and N c a n ideal of N. d ) b r i n q s ( f o r c e r t a i n c l a s s e s o f oroups) answers t o t h e q u e s t i o n s , which n r . ' s ( n r . ' s w i t h i d e n t i t y , . . . ) a r e d e f i n a b l e on a q i v e n a d d i t i v e o r o u p . F o r i n s t a n c e , e v e r y fir. w i t h i d e n t i t y on a c y c l i c o r o u p i s a commutative r i n n . S e v e r a l g r o u p s a r e e x p l o r e d w h i c h c a n n o t be t h e a d d i t i v e o r o u p o f a nr. w i t h i d e n t i t y .

no)

We g o o n b y d i s c u s s i n g o r d e r e d n r . ' s i n e ) ( a n d d i s c o v e r t h a t v e r y few n r . ' s c a n be f u l l y o r d e r e d ) . R e g u l a r n r . ' s a r e s t u d i e d i n f ) , tame n r . ' s i n g ) , w h i l e h ) c o n t a i n s i n f o r m a t i o n o n MS(r), where S i s n o t a f i x e d - p o i n t - f r e e automorphism group. We c i o s e w i t h t h e c o n n e c t i o n s b e t w e e n n r . ' s a n d a u t o m a t a i n i ) a n d a survey on o t h e r tonics i n j ) .

288

$ 9 MORE CLASSES OF NEAR-RINGS

a)

IFP

-

NEAR-RINGS

I n r i n g t h e o r y , t h e f o l l o w i n s two t h e o r e m s a r e c e r t a i n l y among t h e m o s t f a m o u s c o m m u t a t i v i t y t h e o r e m s ( s e e e . q . ( P r o c e s i ) ) : T H E O R E M 1: L e t R b e a r i n o w i t h

!X E R 3

n ( x ) c D W l ~ :x ~ ( ~ x) =

Then l? i s c o m m u t a t i v e . ---T H E O R E M 2 :

L e t R be a r i n g w i t h x , y ~ R3 n ( x , y ) E I N \ C l J : ( x y - y x ) n ( x , v !

= x y - yx.

Then R i s c o m m u t a t i v e . ( T h e f i r s t o n e w a s o b t a i n e d by N. J a c o b s o n ; t h e s e c o n d o n e i s due t o I.N. H e r s t e i n . ) We w i l l g e n e r a l i z e t h e s e r e s u l t s t o c L r t s i n c l a s s e s o f n e a r r i n g s ( i n c l u d i n s t h e d a n r . ' ~ )u s i n o s u b d i r e c t d e c o m p o s i t i c n s . I n o r d e r t o g e t a s a t i s f a c t o r y t r e a t m e n t we s t a r t w i t h a m o r e general c l a s s of near-ri nss:

1.1 lFP-!iEAR-RINGS

9 . 1 DEF1:IITIOti -

A n r . :t i s s a i d t o f u l f i l l t h e i n s e r t i o n - o f -

factors-property

(I F P ) provided t h a t

a , b , n & N : ( a b = 0 ->

anb = 0).

N h a s t h e s t r o n r i I F P i f e v e r y homornorphic i m a o e o f I

11 h a s

the IFP. The n e x t i s a n i n t r i n i s t i c c h a r a c t e r i z a t i o n o f t h e s t r o n q I F P :

289

9 a IFP-near-rings

9.2

PROPOSITION ( P l a s s e r ( 1 ) ) . N h a s t h e s t r o n a I F P : IsN V a , b , n ~ N : ( a b E I =-> a n b c I ) .

v

The p r o o f i s s t r a i a h t f o r w a r d a n d h e n c e o m i t t e d . We w i l l s o o n q e t e x a m p l e s o f I F P - n e a r - r i n n s . characterize these near-rinas.

9.3

B u t b e f o r e we

PROPOSITION ( B e l l ( l ) , P l a s s e r ( 1 ) ) . The f o l l o w i n q assertions are equivalent: ( a ) N has t h e IFP-property. (b)

'd

(c)

ncN:

(0:n) 9 N.

S5i.I:

(0:s) A

N.

Again, t h e proof i s obvious. Observe t h a t e v e r y I F P - n e a r - r i n q N w i t h l e f t i d e n t i t y e i s i n eO = 0 i m p l i e s t h a t e n 0 = 0 , whence n O = 0 f o r a l l nEN.

q ,f o r 9.4

DEFINITION C o n s i d e r t h e f o l l o w i n a p r o p e r t i e s :

(PI):

(Po)

(P2):

W NE

9.5

3

XEN

(Po):

n ( x ) > l : xn(x)

and N i s

=

x.

EQ.

3 n(x,y)>l: ?lo. X,YEN

(P3):

x , y , z c N : xyz = x z y

(P4):

x,yEN

(xy-yx) n ( x y y )

ISN: X Y E I ->

=

xy-yx

and

("weak c o m m u t a t i v i t v " ) . YXEI.

REMARKS ( a ) The ' ' x n ( x ) = x " - p r o p e r t y d o e s n o t i m p l y t h a t Ncno, f o r e v e r y Ncnc f u l f i l l s i t . N r . ' s w i t h ( P l ) a r e c a l l e d " L - n e a r - r i n a s " i n L i a h ( 1 1 ) . 5 e e S z e t o (6), ( 8 ) f o r a characterization v i a sheaf representations. N with xcil: x 2 = x a n d ( P 3 ) s t u d i e d b y R d t l i f f ( 1 ) an: Subr;hnanyay ( 1 )

( b ) Abelisn n r . ' s

were


290

("Boolean s e m i r i n q s " ) . Abelian n r . ' s w i t h (P,) a r e c a l l e d " s e m i r i n o s " t h e r e . The n r . ' s I{ w i t h 1 x c N : x T x a n d ( P 3 ) a r e t h e "8-near-rinos" of Ligh ( 1 4 ) .

(c) ( P 4 ) was c o n s i d e r e d by B e l l (11, ( 2 ) a n d P l a s s e r ( 1 ) . Every nr. w i t h ( P 4 ) i s in B u t , on t h e o t h e r hand, every c o n s t a n t nr. has ( P 3 ) .

no.

9.6

PROPOSITION (Re11 ( 2 ) , ( a ) ( P I ) =>

(5) =>

( b ) Each o n e o f

Lioh (16)).

(Pa). to

(P1)

(P4)

implies the strong IFP-

property. P r o o f . ( a ) : ( P 1 ) -> ( P 2 ) i s i m m e d i a t e . Assume ( P 2 ) a n d X Y E I . Then y x - x y 5 yx ( m o d I ) a n d 3 n c I N \ { l I : y x - x y = ( ~ x - x y )z ~ ( Y X ) ~= y x y x . . . y x : 0 (mod I ) . Hence Y X E I . ( b ) : S i n c e ( P l ) - (P,) a r t i n h e r i t e d t o homomorphic i n a g e s i t s u f f i c e s t o show t h e I F P - p r o p e r t y i n t h i s

c a s e . By ( a ) , we o n l y h a v e t o l o o k a t

(P3): I f

ab = 0

and

( P 4 ) : I f a b s I and whence anbcI

nEN

nEN

by

then

then (P4).

(Pg)

a n b = abn

bas1,

hence

and

(P4).

!In = 0 . b(an)cI,

See e . g . Liph ( 1 6 ) and ( T h i r r i n ) f o r t h e c o n n e c t i o n t o " L u L r i n o s " ( i . e . r i n n s , i n w h i c h e v e r y o n e - s i d e d i d e a l i s twos i d e d ) . C l e a r l y each duo r i n ? i s a stronn I F P - n r . ( b u t n o t c o n v e r s e l y ) , For a d e t a i l e d s t u d y o f " d u o - n e a r - r i n a s " s e e Choudhari ( I ) , ch. V I I I , Choudhari-coyal ( I ) a n d Parnakotaiah-Rao ( I ) . For e a s y r e f e r e n c e , i t rewards t o d e f i n e f o r t h i s chaptet. 9.7

D E F I N I T I O N L e t a n r . H be o f

type I t ype I1

if

N s i m p l e and s i r o n a l y uniform ( 8 . 9 9 )

FIE%,

i f ;icng i s n o t s i r i p l e , b u t t ' l e i n t e r s e c t i o n o f a l l n o n - z e r o ; d e a l s c o n t a i n s no n o n - z e r o i d e PI r) o t E. n t

.

9a IFP-near-rings

29 1

N$n0, fJgnc

t y p e 1 11 if a n d i f P ( a s a b o v e ) h a s a nonzero idempotent then P = N 0' t y p e I V i f NE?'),. type v i f V X , Y E N : xy = 0 . The s t r u c t u r e o f s t r o n q I F P - n e a r - r i n o s i s o i v e n by 9.8

THEOREM ( i i o h ( 1 6 ) ) . Every s t r o n q I F P - n e a r - r i n q N i s a

subdi r e c t product of subdi r e c t l y i r r e d u c i b l e IFP-near-rinas o f type I , I I , I I I , I V or V .

-P- r o o f . L e t N be t h e s u b d i r e c t p r o d u c t o f some s u b d i r e c t l y i r r e d u c i b l e near-t-inns (1.62(a)).

The

lJi

(icsomc index s e t I )

N i l s h a v e t h e I F P - p r o p e r t y by 9 . 1 .

i s simple, use 9 . 3 t o net ( a ) I f Nismo a n d N i i n t o type I o r type V .

Ni

( b ) Now l e t By 1 . 6 0 ( c ) ,

b e n o t s i m p l e and P be as i n 9 . 7 . tlic% P -f; { O l . Assume t h a t P c o n t a i n s t h e

idempotent e p 0. If 3 x € N i : x e -1 x t h e n 0 xe-xe(0:e) 9 N i , so Pc(O:e), ee(o:e) a n d e = e 2 = 0, a contra-

4

d i c t i o n . Hence e i s a r i o h t i d e n t i t y , c o n t a i n e d i n P , whence P = N i , a c o n t r a d i c t i o n . (c) If

Ni

l o ) $. ( N i ) o

i s neither (0:0)

e v e r y i d e m p o t e n t e $. 0 I f x ~ ( O : 0 ) , x = xeEP,

nor

then As i n ( b ) , i s a rinht identity in PIi. hence P = (0:O) = ( N i ) o .

F?&

Then

ncC(N), and k-1 2 (rn ) = 0, SO

n k x - x n k = mxm'-'= ( I F P ! ) = (mx-xm)n(xm-xn)m.. k- 1 k- 1

XEN:

- xmm k -=l ( m x - x m ) m k - '

0

((mx-xm)m)

=

(m(xm)-(xmjm)

( P 2 ) a o a i n y i e l d s m(xrn)-(xm)m = 0. aSovc, i t tut-r,; G I ; t +-hat ; r : x - x r 1 2 = O , :vhe?r,?

Applyinq AS

nr! =

by t h e I F P .

=

(P2).

by

Now 8ssuc:e t h d t take

so

= 0,

(xn-nx)xn

XEN:

since

nxxn = 0

and

V

Then

mx-xm = 0 .

.

9a I FP-near-rings

29 5

( b ) From ( a ) a n d t h e I F P - p r o p e r t y o n e a e t s ( a s f o r

rinqs) that the s e t Np%(Ni) of N i f o r m s an i d e a l . ( c ) I f N p t ( N i ) = Ni, (P2) commutativity of (N,. ) . N p t ( N i ) = {O}

(d) If

and

of a l l n i l p o t e n t elements

instantly results the

N F ~ , , N i i s inteqral

by 9 . 1 3 ( a ) ,

h e n c e a b e l i a n b y 9 . 1 3 ( b ) , c o n s e q u e n t l y a r i ~ q( 6 . 9 ( c ) a n d 6 . 6 ( c ) ) a n d t h e r e f o r e a c o m m u t a t i v e otie ( T h e o r e m 2 ) . I f Pi&?,, N i s commutative by 9 . 1 3 ( b ) .

-+ f i p t ( V i )

(e) If

$- I J i ,

consider

TTi:

= ''i/Npt(Xi)'

Ri

h a s no n o n - z e r o n i l p o t e n t e l e m e n t s , b u t i s a n a i n dg. w i t h ( P 2 ) . By ( d ) , fli i s a commutative r i n q . So f o r a l l n',n"EN, n ' n " - n " n ' c N p t ( N i ) , from which

we cret

n'n"

= n"n'

by

P2)'

( f ) S i n c e s l i rIi a r e commutative n e a r - r i n q s , same a p p l i e s t o t i . (91

"

N has

c e n t r a l

Anyhow,

(N,t)

ring.

Each a - n r .

(9) ( L i g h ( 1 5 ) ) .

w

Call

i s torsion

N F ) ~

Every a-nr.

-dENd.

ncIN

with

A nr.

b

(N,t).

an a - n r .

if

(Po)

dENd

implies

i s a commhtative r i n o .

(Pg)

N with

: ( x Y ) ~= x n y n .

this property (or with = xn+yn)

and e a c h e l e m e n t

w i t h o u t n i l p o t e n t elements i s a

Each n r .

x,yeN

a l l idempotents dre

N i s subdirect product of

has a s q u a r e - f r e e o r d e r i n

(f) (Liqh (8)).

fulfills

Every a-nr. X,YEN:

n&IN

=

(N,.).

w i t h o u t n i l p o t e n t elements and w i t h

x , y ~ N : ( x Y ) ~= x 2 y 2

( h ) (Liqh-Utuni

N with

(xty)"

has o n l y n i l p o t e n t commutators o f

NcWl

all

i s abelian.

( 1 ) ) . 9 i s a C!-nr.

n E N : nN = nPin

( N n = nNn,

one irnDiies t h e o t h e r :

(C2-nr.)

if

respectively).

Neither

59 MORE CLASSES OF NEAR-RINGS

298

I f F i s a f i e l d then n o t C2-nr. .

MafF(F)

(1.4) i s a C1-,

A f i n i t e i n t e g r a l nr.

has C1,

but not

but

C2.

E v e r y C2-nr. ( b u t n o t e v e r y C 1 - n r . ) h a s t h e I F P . N i s C1- a n d C 2 - n r . i f f N i s C1-nr, a n d e v e r y idempotent i s central. S e e t h i s p a p e r f o r d e c o m p o s i t i o n t h e o r e m s f o r C 1 - and C2-nr.'s with f i n i t e n e s s conditions. ( i ) A r i n g R i s called a P 1- r i no i f for all rrR t h e r e i s a c e n t r a l idempotent ro with r r 0 = r a n d e 2 = r&R : ( e r = r e ==> r 0 e = r * ) ( t h i s P I h a s n o t h i n ? t o d o :.rith o u r ( P 1 ) ) . See P l a s s e r ( 1 ) f o r a s i m i l a r c o n c e p t f o r n e a r - r i n ? s . ( j ) F o r more r e s u l t s s e e B e l l (!I)% L i g h ( l ? ) , F'arin R a m a k o t a i a h - R a o ( 2 ) , ( 5 ) a n d Kim-Park ( 1 ) .

(2),

2 . ) p-NEAR-RINGS L _

I ON L e t p b e a p r i m e . I! n r . --9 . 2 2 D E F I N I T--

r i n q provided t h a t

_ i

Evidently, every p-nr.

~

~ E N :xp

=

has p r o p e r t y

x

PI i s c a l l e d a ---p-nearA px =

0.

(Po).

9.23 PROPOSITIOIJ ( P l a s s e r ( 1 ) ) . A p - n r .

w i t h l e f t identity i s

zero-symnetric. P r o o f . L e t e be t h e l e f t i d e n t i t y . T h e n i t i s e a s i l y shown by i n d u c t i o n t h a t X E N k j kcM : ( e + x O ) k = e + k ( x 0 ) . Hence e t x O = ( e + x O ) P = e + p ( x O ) :: e , whence x0 = 0 . 9.24 REMARK ( P l a s s e r ( 1 ) ) . 9.23 does not h o l d f o r yenera1 o r . ' s with (P,).

9.25

CORQLLARY ( R a t l i f f ( 1 ) ) . A p - n r .

with (P3) a n d non-zero d i s t r i b u t i v e e l e ~ e f i t si n ever;; h o m c n 3 r ~ k i c i n a p e i s i s c morphic t o a s t i b d i r e c t p r o d u c t of c o p i e s of t h e f i e l d 8,, h e n c e a ring.

9a IFP-near-rings

299

-Proof.

By 9 . 2 0 , N i s a s u b d i r e c t p r o d u c t o f s i m p l e Ni w i t h i d e n t i t y . Rino t h e o r y commutative p-rinqs

t e l l s us t h a t

%

Ni =2,.

9.26 THEOREM (Plasser (1)). A f i n i t e p-nf. the field

PI i s i s o m o r p h i c t o

ZP.

-.P r o o f . N i s a D i c k s o n n f . , f o r N c a n n o t b e o n e o f t h e 7 e x c e p t i o n a l c a s e s ( 8 . 3 4 and t h e s u b s e q u e n t d i s c u s s i o n ) , s i n c e i n each one o f t h e s e c a s e s A 5 = A , b u t Ei5 R.

n

q E p 3 nEIN : IIJI = q b v 8.13. Since ( N , + ) i s a f i n i t e p-qroup, / N I i s some power o f p , c o r ~ s e q u e n t l y q = p . Now ( N " , . ) h a s q e n e r a t o r s a , b w i t h b - 1a b = a q = = ap ( 8 . 3 3 ) . Thus a b = b a P = h a , N i s commutative !OW

3

(P3). and h e n c e h a s Now t h e r e s u l t f o l l o w s f r o m 9 . 2 5 .

9-27 REMARKS ----___ ( a ) Cf. a l s o 8 . 3 5 . ( b ) The f i n i t e n e s s c o n d i t i o n i n 9 . 2 6 i s i n d i s p e n s a b l e , f o r

there exist infinite p-fields.

A n a p p l i c a t i o n o f 2 . 5 2 ( b ) n i v e s w i t h 9 . 1 7 and 9 . 2 6 t h e f o l l o w i n n 9 . 2 8 C O R O L L A R Y ( P l a s s e r ( 1 ) ) . L e t :i be a f i n i t e p - n r . w i t h I F P and w i t h s o n - z e r o d i s t r i b u t i v e elein.ents i n e v e r y nonz e r o homomorphic i m a o e . Then 14 i s i s o m o r p h i c t o a ( f i n i t e ) hence a f i n i t e p - r i n o . d i r e c t sum o f c o p i e s o f zP ' 9.29 R E M A R K R a t l i f f ( 1 ) s t u d i e d p - n r . ' s

N ( e s p e c i a l l y Tor

p=3

a n d P = 5 ) , w h i c h c a n be d e r i , J e d f r o m a p - r i n c l R i n a way that (N,+) = ( R , + ) and the'product i n N i s defined v i a a f i x e d p o l y n o m i a l f u n c t i o n o v e r R . The n r . ' s c o n s i d e r e d

i n this dissertation fulfill

(P,)

and

(P3).


300

3.)

B O O L E A N NEAR-RINGS

I t d o e s n o t seem t o be q u i t e c l e a r how t o d e f i n e a B o o l e a n n e a r r i n g . S o we t a k e w h a t s e e m s t o be t h e m o s t q e n e r a l p o s s i b l e d e f i n i t i on.

9.30 DEFINITION A nr. N i s B-o o l e a n : --

XCH : x2 = x.

Hence a B o o l e a n n r . i s a ( P o ) - n e a r - r i n q w i t h all x.

n(x) = 2

for

9 . 3 1 R E M~ARKS

( a ) Every c o n s t a n t nr. i s a Boolean nr. w i t h (Pj), h u t

(Po)

and

not a 2-nr. i n g e n e r a l .

( b ) A Boolean nr. w i t h

and non-zero d f s t r i b u t i v e e l e m e n t s i n e v e r y n o n - z e r o homomorphic image i s a s u b d i r e c t p r o d u c t o f c o p i e s o f iZ2. This r e s u l t o f Ligh ( 1 4 ) f o l l o w s from 9 . 2 5 .

( c ) (Ligh ( 5 ) , (14),

(P3)

(a 6-nr.)

( 8 ) , ( l o ) , H e a t h e r l y ( 7 ) ) . The same

a s s e r t i o n h o l d s f o r d q . B o o l e a n n r . l s . Clf c o u r s e , t h i s f o l l o w s from 9 . 1 8 , b u t t h e r e i s a l s o a d i r e c t e l e m e n t a r y proof i n Liqh ( 1 0 ) . ( d ) See p . 4 1 8 / 4 1 0 f o r a l i s t o f a l l B o o l e a n n r . I s d e f i n a b l e on t h e two n o n - a b e l i a n a r o u p s o f o r d e r 8 .

( e ) F e r r e r o - C o t t i ( 2 ) , ( 3 ) c o n s i d e r e d nr.'s w i t h t h e i d e n t i t i e s abc = acbc = a b a c . These a r e t h o s e ones w h i c h c o n t a i n a n i d e a l I w i t h I' = I 0 1 a n d N / I is a Boolean r i n q . ( f ) A Boolean n r . w i t h l e f t i d e n t i t y i s a Boolean r i n g with

i d e n t i t y (Ligh ( 5 ) ) . ( 9 ) More r e s u l t s a r e c o n t a i n e d

Ramakotaiah-Rao ( 2 ) .

i n Heatherly-Stone

( 1 ) and

301

9b Near-rings without

9 . 3 2 C O R O L L A R Y ( H e a t h e r l y ( 7 ) ) . A Boolean n r . i s a f i n i t e d i r e c t sum o f i d e a l s which a r e

.no

n r . ' s with the t r i v i a l ~ u l t i n l i c a t i o n xy =

with DCCI inteoral simple

[

x

Y 9 0

O

y = o

P r o o f . Apply 9 . 1 0 , 2 . 5 2 ( b ) a n d t h e f a c t t h a t e v e r y n o n zero element i s (as a n idempotent s e e t h e proof o f 9.13(b)) a right identity.

-

9 . 3 3 EXAMPLES

( a ) ( C l ay - La wve r ( I ) ) . Le t

be a Boolean r i n o w i t h i d e n t i t y 1. Le t a ' : = a t 1 and a v b : = ( a ' A b ' ) ' . I f XEB, define for a,bEB a b : = a A ( b v x ) . Then (B,+,ax) i s a Roolean n r . w i t h (Pj) which i s a r i n q i f f x = 0. N r . ' s d e r i v e d from Boolean r i n g s a r e c a l l e d " s p e c i a l B o o l e a n n e a r - r i n n s " i n t h i s p a p e r . Their- i d e a l s t r u c ture i s considered. (B,+,A)

Q~

I _

_ _ _ I _ _ _ I -

( b ) Subrahmanyam ( 1 ) c a l l e d a n a b e l i a n Boolean n r . w i t h

(P3)

"Bool e a n s e m i r i n n " . Every P 1 - r i n g ( 9 . 2 1 ( i ) ) ( B , t , = ) g i v e s r i s e t o a Boolean s e m i r i n a ( B , t , * ) , where a s b : = ab 0 Every c o n s t a n t a b e l i a n n r . i s a Boolean s e m i r i n a . A Boolean s e m i r i n g can be r e p r e s e n t e d as a d i s j o i n t u n i o n of " n e a r l y d i s t r i b u t i v e " l a t t i c e s . S ee t h i s p a p e r f o r more d e t a i l s .

.

b ) NEAR-RIPiGS WITHOUT

1. \ N E A R - R I N G S l.IITtiC)UT N I L P O T E N T E L E M E N T S

N r . ' s w i t h o u t n o n - z e r o n i l p o t e n t e l e m e n t s came u p a t s e v e r a l d i f f e r e n t p l a c e s i n o u r d i s c u s s i o n o f n e a r - r i n o s . We c o l l e c t some of t h e r e s u l t s c o n c e r n i n g t h e s e n e a r - r i n o s .


302

9.34 -

REMARKS Let N be a n r . w i t h o u t n o n - z e r o n i l p o t e n t e l e m e n t s . Then ( a ) N h a s no n i l ( p o t e n t ) s u b s e t s ( 2 . 9 6 ) .

NEW,

has D C C N t h e n e v e r y non-zero N-subgrou? c o n t a i n s a non-zero idempotent ( 3 . 5 1 ) . Moreover, i n this case = Jo(N) = V(N) = = I01 (5.40).

(b) If

P(N)

( c ) I n any c a s e ,

T(N) = p(X) =

{01.

9. 3 5 E X A M P L ES ( a ) Every c o n s t a n t n r . has n o non-zero i i i l c o t e n t e l e m e n t s . ( b ) Every i n t e g r a l n r .

(hence e v e r y n f . ) has t h i s p r o p e r t y ,

too. The c o n n e c t i o n t o t h e p r e v i o u s c h a p t e r i s g i v e n by 9 . 3-6 - T H-E O- R E M ( B e l l ( l ) , M a r i n ( l ) , Q a r n a k o t a i a h - ' J a o ( 2 ) ) ~ Let N be zero-symmetric.

Equivalent are:

( a ) N has no non-zero n i l p o t e n t e l e m e n t . ( b ) N i s a subdirect product of interlral n r . ' s .

~Proof. ( a )

( b ) i s nothinn e l s e trran i n t h e proof o f 9 . 1 3 ( a ) : N has a family o f i d e a l s I x ( ~ E N " ) w i t h z e r o i n t e r s e c t i o n and each N / I x i s integral. =>

( a ) : I f x " = 0 , i n e a c h component n i ( N ) o f t h e s u b d i r e c t r e p r e s e n t a t i o n o f N we a e t i ~ ~ ( x ) = Q , whence x = 0 . ( b ) =>

Hence w e w i l l d e v o t e t h e n e x t nur?ber t o i n t e o r a l n e a r - r i n a s . B u t b e f o r e , some more r e s u l t s m i n h t be a p p r o p r i a t e . 9 . 3 7 PROPOSITION(Bel1 ( l ) , H e a t h e r l y ( 7 ) , M a r i n (l), R a m a k o t a i a h Rao ( 2 ) ) . A n r . N E % 0 w i t h o u t n o n - z e r o n i l p o t e n t e l e m e n t s i s an IFP-nr.


xy = 0

If

Proof.

(x,y~N)

so

( y x 1 2 = 0,

= n ( y x ) = nO = 0,

9.38

then

nr.

nEN:

xny

whence

(ny)x =

s o N has t h e IFP.

COROLLARY ( H e a t h e r l y ( 7 ) ) .

Every s u b d i r e c t l y i r r e d u c i b l e

w i t h o u t non-zero n i l p o t e n t s i s i n t e o r a l .

riEQ

non-zero

y x y x = yOx = 0,

ow

0.

yx

303

Every

idempotent i s a r i o h t i d e n t i t y .

P r o o f . The f i r s t a s s e r t i o n h o l d s by 9 . 1 3 ( a ) .

$. 0

e

If

whence

XEN:

i s idempotent,

(xe-x)e = 0,

xe = x.

To 9 e t m o r e , we h a v e t o i n p o s e s o m e f i n i t e n e s s c o n d i t i o n s o n P I . 9 . 3 9 P R O P OS ITION

( H e a t h e r l y ( 7 ) ) . Let

irreducible nr.

(a)

(b)

N N

be a s u b d i r e c t l y

w i t h DCCN and w i t h o u t non-zero

{O}

n i l p o t e n t elements.

tlE'TIo

Then

i s i n t e q r a l and 2 - p r i m i t i v e on

N.

has a r i g h t i d e n t i t y .

( c ) l i d =f. 0

==,

N

( d ) I f N i s dg,

-Proof.

i s a nf.

.

t h e n ti i s a f i e l d .

( a ) Consider,

T h e r e i s some

XEN*,

for

... .

nE.DJ w i t h

i n t e g r a l by 9.36,

so

3

t h e c h a i n YCx?Nx2?Plx 3. n+l P! i s fixn Nx

Nxn = ( N x ) x n

T h e r e f o r e Pi i s 2 - p r i m i t i v e o n (b) h o l d s by 4.46

implies

..

PI = N x .

N.

o r by 9.38.

( c ) By t h e same a r g u m e n t a s i n t h e p r o o f o f 3 . 1 3 ( b ) ,

N

Now a p p l y 4 . 4 7 ( a )

c o n t a i n s an i d e n t i t y .

and 9 . 1 7 .

( d ) i s obvious, 9 . 4 0 REMARK ( H e a t h e r l y ( 7 ) ) . abelian nr.'s elements,

N

with

There e x i s t even f i n i t e simple (PI)

and w i t h o u t non-zero n i l p o t e n t

which are not n f . ' s .

We c a n r e d u c e t h e t h e o r y o f n e a r - r i n a s w i t h D C C E l i r n d n o n o n z e r o n i l D o t e n t e l e m e n t s t o t h a t ~f

9.39:


304

9 . 4 1 THEOREM ( H e a t h e r l y ( 7 ) ) . zero n i l p o t e n t elements. 2-semisimple

Let

NEnO

Then

N

h a v e DCCN a n d n o n o n -

has a r i r l h t i d e n t i t y ,

a n d t h e f i n i t e d i r e c t sum o f n r . ' s

f u l f i l l a l l c o n d i t i o n s o f 9.39.

is

which

If e v e r y n o n - z e r o homo-

Pi h a s n o n - z e r o d i s t r i b u t i v e e l e m e n t s t h e n N i s a f i n i t e d i r e c t sum o f n f . l s ; i f N i s d q . t h e n Pi i s a

morphic imaqe of fi'nite

d i r e c t sum o f f i e l d s .

P r o o f . Decompose nr.ls

Pii

N into subdirectly irreducible inteqral ( 9 . 3 6 ) . I n f a c t , N i s a f i n i t e d i r e c t sum

o f t h e s e ones

Now a p p l y 9 . 3 9 ( b ) ,

(2.52(b)).

9.39(a)

and 5.49.

9 . 4 2 C0RrJLLAP.V ( t l e a t h e r l y ( 7 ) ) .

If

i s a f i n i t e nr.

NFno

w i t h o u t non-zero n i l p o t e n t elements. T h i s i s c l e a r by 9 . 3 8 that

Cf.

n(x)

N

Then

has

(PI).

a n d 9 . 4 1 ( I l e a t h e r l y a o e s on t o show

can be chosen t o be c o n s t a n t f o r a l l xzN.

(9.4)

a l s o L i g h (11)).

Moreover,

we h a v e some i n f o r m a t i o n c o n c e r n i n q t h e n e a r - r i n o ;

i n d i s c u s s i o n , which guys belono t o t h e c e n t e r

( I ) ) . Let

9 . 4 3 PROPOSITION ( B e l l potent elements.

o f N:

C(N)

have no Eon-zero n i l -

FIE'Q,

Then

(a) Every d i s t r i b u t i v e idempotent i s c e n t r a l . all

(b) I f Proof.

idempotents ?re i n

C(K).

F i r s t we s h o w t h a t f o r e a c h i d e m p o t e n t e ,

XEN: e x = e x e . rl

e(ex-exe)

Now

and

(-exe).(ex-exe)

(ex-exe)e

ex(ex-exe)

= ( - e x ) O = 0.

= ex(ex-exe)t(-exe)(ex-exe)

= '3,

= 0

so (9.37)

(IFP).

Therefore

= O t 0 = 0,

Hence (ex-exe!*

whence

e x - e x e = 0.

(a) If

ecNd,

= exe-exe

= 0,

xe = exe = ex.

v

XEN: e(xe-exe) hence

(xe-exe!e

= exete(-exe) = 9,

whence

=

305

9 b Near-rings without

( b ) I f N h a s a n i d e n t i t y 1 , c o n s i d e r a g a i n some i d e m p o t e n t e . ( 1 - e ) e = 0 , s o \1 xcN: ( l - e ) x e = 0 . Also, ( x e - e x e ) e = xe-exe and (1-e)xe = x e - e x e , therefore (xe-exe)' = (xe-exe)e(xe-exe) = = (xe-exe)(l-e)xe = 0 , so xe = exe = ex f o r a l l XEN.

9 . 4 4 REMARKS ( a ) See Marin ( 1 ) f o r c h a r a c t e r i z a t i o n s o f t h o s e n e a r r i n g s w i t h o u t non-zero n i l p o t e n t e l e m e n t s which a r e ( f i n i t e l y or n o t ) c o n p l e t e l y r e d u c i b l e i n t o c e r t a i n o t h e r n e a r - r i n a s . See a l s o Szeto-Nong ( 1 ) . ( b ) Recall 9 . 2 1 ( f ) .

( c ) A g a i n , l e t N ~ 9 j ' ~ h a v e no n o n - z e r o n i l p o t e n t e l e m e n t s . Then ( B e l l - L i g h ( I ) ) : a ) I f N i s d g . w i t h f i n i t e l y many s u b n e a r - r i n o s ,

N i s

a f i n i t e commutativc rinq.

B ) I f 14 h a s a t m o s t 2 i d e n p o t e n t s a n d no p r o p e r ( f i n i t e l y many) s u b n e a r - r i n q s , N i s a f i n i t e f i e l d (a near-fi el d , respectively). ( d ) Don't forget t o observe 9 . 5 4 .

2 . ) N E A R - R I N G S !.IITHOUT Z E R O D I V I S Q R S ( I P J T E G R A L N E A R - R I ! i G S )

I

9 . 4 5 EXAMPLES

( a ) Every c o n s t a n t nr. i s i n t e c r r a l .

d e f i n e s a n i n t e g r a l nr.

(r,+,*)

(cf.

1.4(b)).

306


So o n e c a n s a y n o t h i n c l a b o u t t h e a d d i t i v e a r o u p o f a n i n t e q r a l n e a r - r i n g . To o v e r c o m e t h i s we w i l l g i v e t h e f o l l o w i n q

9 . 4 6 DEFINITION A n i n t e g r a l n r . N i s n o n - t r i v i a l i f i t s m u l t i p l i c a t i o n i s not one of 9 . 4 5 ( a ) o r ( b ) . These n o n - t r i v i a l i n t e g r a l n r . ' s a r e sometimes c a l l e d " n e a r i n t e g r a l d o m a i n s " ( s e e L i g h (131, H e a t h e r l y - O l i v i e r ( l ) , ( 2 ) , Adams ( l ) , ( 2 ) ) . B u t t h e y a r e n o t a l w a y s e m b e d d a b l e i n t o a n e a r - f i e l d , s o we r e s e r v e t h i s d i s t i n q u i s h i n q name t o a more s p e c i a l c l a s s o f n o n - t r i v i a l i n t e g r a l n e a r - r i n a s ( s e e 9.52 and 9 . 6 5 ) . See a l s o O l i v i e r ( 2 ) . 9 . 4 7 PROPOSITION ( C l a y ( a ) , H e a t h e r l y - O l i v i e r I f N i s i n t e g r a l t h e n NEQ o r N~fl,.

( l ) , Plasser (1)).

P r o o f . S u p p o s e t h a t 3 X E N : x0 ;f; 0 . T h e n f o r a l l n E N h a v e (nxO-n)xO = n x 0 - n x 0 = 0 , w h e n c e n x O = n . Hence nO = nxOO = nxO = n , a n d N i s c o n s t a n t .

we

T h u s every n o n - t r i v i a l i n t e g r a l near-ring i s zero-symmetric. Integral near-rinqs also appear in previous chapters. I n order t o p r e s e n t a good a e r i a l v i e w on t h i s t o p i c we c o m p i l e t h e s e facts: 9 . 4 8 R E M A R K S R e c a l l t h a t an i n t e y r a l n e a r - r i n a N h a s t h e followinq properties: ( a ) N has t h e r i g h t c a n c e l l a t i o n law ( l . l l l ( a ) ) , ( b ) I f N i s f i n i t e and n o n - a b e l i a n t h e n e a c h e l e m e n t o f

N has a u n i q u e s q u a r e - r o o t ( 1 . 1 1 2 ) . ( c ) N i s a prime near-rincj ( 2 . 6 6 ) . ( d ) I f N i s n o n - t r i v i a l i n the s e n s e o f 9.46 and h a s the DCCI t h e n N i s s u b d i r e c t l y i r r e d u c i b l e ( 2 . 1 0 7 ) . A p p l y i n g 9 . 3 9 we q e t : I f PI h a s m o r e o v e r t h e D C C N t h e n t h e r e e x i s t s a r i a h t i d e n t i t y , !id A implies t h a t N i s a n f . (Ligh-Malone ( l ) ) ,N i s d q . i m p l i e s t h a t N i s a f i e l d ( c f . a l s o 6 . 1 4 ( b ) ) , N i s 2 - o r i m i t i v e 9~ N

307


a n d s o N i s s i m p l e ( H e a t h e r l y ( 7 ) ) . See a l s o G r a v e s -

Malone ( 1 ) . ( e ) On t h e w h o l e , 9 b ) l ) i s a p p l i c a b l e , f o r N h a s n o nonzero n i l p o t e n t elements. So i f N i s n o n - t r i v i a l , i t has the I F P .

9 . 4 9 R E M A R K t o 9 . 4 8 ( d ) . Llithout c h a i n c o n d i t i o n s one c a n n o t conclude t h a t a n i n t e q r a l n r . N with N d f {O} i s a near(7.78). N i s f i e l d : t a k e a f i e l d F a n d form N : = Fo[x] i n t e g r a l ( 7 . 6 8 ( c ) , l . l l l ( a ) ) , each a x ( a E r I ) i s in lid, b u t N i s n o n f . ( 7 . 6 8 ( b ) ) . I n f a c t , f o r e v e r y kcIN,

k + l ~ ~ + l t . . . [ a ~ ,, .a. .~E +F I ~ i s a n i d e a l a n d 11=12=13= i s a s t r i c t l y descending chain (Heatherly ( 7 ) ) . C f . a l s o Graves-Malone ( 1 ) .

Ik:

= { a k xk +a

...

9 . 5 0 T H E O R E M ( F e r r e r o ( 8 ) ) . L e t N be a f i n i t e i n t e q r a l n e a r - r i n ? N i s n o n - t r i v i a l N i s p l a n a r . P r o o f . -->: C o n s i d e r 6 o f 8 . 9 6 f b ) . S i n c e ncN*: Nn = N ( 9 . 4 8 ( d ) ) . Each g a (aEN") i s a monomorphism s i n c e N i s i n t e g r a l . N i s f i n i t e , so I; I A u t ( N , t ) . N i s n o n - t r i v i a l , so 6 {id]. G i s also fixed-point-free (Heatherly-Olivier ( 1 ) ) : Let g a (aEN*) have a f i x e d - p o i n t no 0. Let x be a r b i t r a r y i n N . S i n c e N n o = N , 3 yxcN: x = y x n o

+

Hence p a ( x ) = x a = y x n o a = y x c l a ( n o ) = y x n o = x , so g a = i d . S i n c e N i s t r i v i a l l y s t r o n q l y uniforni we may a p p l y 8.100 a n d a r e t h r o u g h . I y e r : (4; = a)

finite

-Proof.

=

v ($;EAut(r,+)).

(a) f o l l o w s from t h e f a c t t h a t

(To,+)

( b ) and ( c ) r e s u l t from c o n s i d e r i n o 9 . 1 0 2 THEOREM ( H e a t h e r l y ( 1 ) , ( 2 ) ) . on the f i n i t e simple qroup

r

Let (r,t).

11 ( r , t ) .

Ker b;.

= (r,t,.)

Then

r

be a n r . f a l l s i n t o one

o f the following d i s j o i n t classes: i s the "zero m u l t i p l i -

( i n t h i s case,

( a ) v y c r : 4; cation").

= 6

( b ) I'd = { o l

a n d I' h a s a r i g h t i d e n t i t y . and r has an i d e n t i t y .

(c) r d

=+

Io)

P r o o f . Suppose t h a t i s not the zero multiplication. 9.101(c), 3 6 e r : @:: l' + r E A u t ( r , + ) . NOW

3

* k k e n : (4,) = idr

,

= ya If

and

rd 8 I o ) ,

Since

ak

Y

.\r

+

Ya

By

* k

(0,) (Y)

yer : y = i d ( y ) =

=

i s a right identity. take

6cri.

Consider

,Ilr

i s not the zero multiplication,

(o:r) 4 (I-,+-),

:r+r EEnd(r,t). y+ay

(o:r) = {ol

6 Q $. 6 a n d ( a s a b o v e ) & $ ~ A u t ( r , t ) a n d some p o w e r o f 6 i s a l e f t i d e n t i t y , hence the identity. for

Observe t h a t 9.109

Hence

i m p l i e s t h a t i n c a s e ( c ) o f 9.102,I'

be a f i n i t e prime f i e l d . We now t u r n t o c y c l i c g r o u p s .

has t o


326

9 . 1 0 3 THEOREM ( H e a t h e r l y ( 1 ) a n d o t h e r s ) .

B with a generator

Bn . o r ring.

then N i s a commutative

gEPid

N i s an a b e l i a n dpnr., hence a r i n q .

I n t h i s case,

Proof.

L e t N be a n r . on

E v e r y r i n g on a c y c l i c g r o u p i s c o m m u t a t i v e ( s e e (Beaumont)). 9 . 1 0 4 COROLLARY ( H e a t h e r l y ( 1 ) ) . I f N i s a n r . or Z

with

(01

Nd

t h e r e i s some

Pp

(usual product i n Proof.

If

with

xEN

dcNd

on

zp

(PEP

1

t h e n N i s a commutative r i n q and or

$. 0 ,

i s

shows t h a t 1 i s a l s o

n,n'EN

: nn'

= n-n'*,x

Z ). a s h o r t c a l c u l a t i o n (cancel d!)

EN^;

--

now we may a p p l y 9 . 1 0 3

t o get the f i r s t assertion. Let

1.1 = : x

and

n n ' = (l+ , . . +l ) ( l t n-summands 9.105

n,n'EN.

. . .t l )

1cNd.

= n*n'*x.

n'-summands

R E M A R K T h e same r e s u l t a s i n 9 . 1 0 4

if

Then

= n.n'*(l.l)

holds i n every

Zn On t h e o t h e r h a n d , H e a t h e r l y ( 1 ) q i v e s a n

E4 which i s n o t a r i n n ( i n f a c t , 1 and 3 a r e n o t d i s t r i b u t i v e ) .

example o f a n r . o n

For t h e next r e s u l t , l e t C(k,j) be t h e number o f c o m b i n a t i o n s o f k e l e m e n t s t o t h e c l a s s j ) . N i t h o u t p r o o f we s t a t e 9 . 1 0 6 THEOREM ( R . J a c o b s o n ( 1 ) ) . T h e n u m b e r o f d i f f e r e n t d e f i n a b l e on

(Zp.+)

nr.'s

( P E P ) i s given by

Rore i n f o r m a t i o n s can be f o u n d i n A d l e r ( l ) , F e i g e l s t o c k ( 2 ) and Heatherly (2).

9d Near-rings on given groups

327

3 . ) NEAR-RINGS WITH IDENTITIES O H G I V E ? { GROUPS -

ble s t a r t w i t h

9 . 1 0 7 PRO-P O S I T I O N ( C l a y ( 4 ) ) . L e t N = ( r , + , * ) be a n r . on m Then ~ E N i s a n i d e n t i t y o f N i f f m l = i d arid

w

YET:

OY(U

r,.

= Y.

The p r o o f i s o b v i o u s .

O u t o f 9 . 1 0 3 a n d 9 . 1 0 4 we q e t ( o b s e r v e t h a t u n d e r t h e a i v e n assumptions, x o f 9.104 i s i n v e r t i b l e i n :I): 9 . 1 0 8 C O R O L L A R Y ( C l a y - M a l o n e ( 1 ) ) . I f N i s a n r . E??, on t h e c y c l i c q r o u p (N,+) t h e n 11 i s a c o m m u t a t i v e r i n ? . A l l n r . ' s on ( N , t ) a r e i s o m o r p h i c . T h e r e a r e o ( n ) ( @ t h e E u l e r f u n c t i o n ) o n e s on ( Z n , t ) a n d 2 on Z.

9.109

C O R O L L A__ R Y There a r e e x a c t l y p - 1 n r . ' s with i d e n t i t y d e f i n a b l e on ( h p , t ) ; a l l o f t h e m a r e i s o m o r p h i c t o t h e f i e l d Zp and hence a l l a r e f i n i t e prime f i e l d s .

T h i s r e s u l t was o b t a i n e d by M a l o n e , C l a y , Maxson a n d H e a t h e r l y under d i f f e r e n t circumstances. O b s e r v e t h a t i f i n ( r , t , - ) ~ ? ' ?( ~r , t ) i s a b e l i a n w i t h e x a c t l y ?r o n e p r o p e r s u b o r o u p t h e n ( r , + ) = Zp2 a n d (I-,+,.) is a c o m m u t a t i v e r i n r : by 9 . 1 0 8 ( L i q h ( 9 ) ) . B u t t h e r e do e x i s t n o n r i n g s w i t h i d e n t i t y on rrroups o f o r d e r p 2 ( c f . a l s o 9 . 1 1 5 ( c ) ) : 9 . 1 1 0 PROPOSITION (Maxson ( 1 ) ) . F o r e a c h P E P t h e r e e x i s t s a g r o u p r o f o r d e r p 2 a n d a non-rice w i t h i d e n t i t y on r .

T h e p r o o f i s e s t a b l i s h e d by d e f i n i n g a m u l t i p l i c a t i o n on (I-,+): = ( H p , + ) B ( Z p , t ) i n an a p p r o p r i a t e manner ( s e e Mzxson ( 1 ) f o r d e t a i l s . )

328


Now we s t u d y n r . ' s o f s q u a r e - f r e e o r d e r . F i r s t we need 9 . 1 1 1 P R O P O S I T I O N ( C l a y - M a l o n e ( l ) , Maxson ( 1 ) ) . L e t (I-,+,*) be a n r . w i t h i d e n t i t y 1 on t h e f i n i t e q r o u p r. Let o r d ( y ) be t h e o r d e r o f Y E r . T h e n o r d ( 1 ) = l . c . m . I o r d ( y ) l y ~ r ) = : II. Proof. I f yEr, o = oy = ( o r d ( l ) . l ) y = o r d ( l ) . y , SO o r d ( y ) / o r d ( l ) . Hence f . / o r d ( l ) . B u t l c r , h e n c e o r d ( l ) / I I whence o r d ( 1 ) = II. 9 . i 1 2 T H E O R E M- ( F l a x s o n ( 1 ) , ( 2 ) ) . s q u a r e - f r e e o r d e r . Then i s a c o m m u t a t i v e rincr. ,-

(I-,+,* j E n 1 have f i n i t e ( r , + ) i s c y c l i c , a n d (I-,+,.) Let

P r o o f . Let = plp 2...pr, where p l , . . . , p r are d i s t i n c t primes. U s i n o t h e Sylow t h e o r y we p e t f o r Hence e a c h i E { l , ..., r ) some y i € r o f o r d e r p i . [GI t ord(1) = l . c . m . { o r d ( y ) l y ~ r l t 2 l.c.m.rord(y,) So o r d ( 1 ) = I G I

,..., o r d ( y r ) )

= (GI.

and G i s c y c l i c . N o w u s e 9 . 1 0 3 .

Several groups cannot b e a r a nr. w i t h i d e n t i t y ( c a l l a s u b s e t P o f a p a r t i a l l y o r d e r e d s e t an a n t i c h a i n i f no d i s t i n c t e l e m e n t s a r e comparable): R E M (Krimmel ( 1 ) , ( 2 ) ) . Let (r,+) be a g r o u p h a v i n r l 9 . 1 1 3 T H E Oe l e m e n t s y l , . . ,y, of d i s t i n c t prime o r d e r s p l , pr ( r 1 2 ) . I f e v e r y a n t i c h a i n i n t h e l a t t i c e of normal subgroups o f r has c a r d i n a l i t y < r then (r,t) c a n n o t be t h e a d d i t i v e q r o u p o f a n r . w i t h i d e n t i t y .

...,

.

P r o o f . S u p p o s e t h a t ( r , + , * ) i s a nr. w i t h i d e n t i t y 1. I f there are i,jc{l, r l w i t h ( o : y i ) F ( o : y J. ) then h: ( r y i , + ) * ( r y j , + ) i s a well-defined aroupYYi * YYi e

...,

homomorphism. B u t

h ( y i ) = h ( l y i ) = l y j = yj,

whence

329

9d Near-rings on given groups pj = ord(yj)/ord(yi)

{ ( o : y l ) ,...,( o : y r ) l a contradiction.

pi, so i = j. Hence i s an a n t i c h a i n w i t h r e l e m e n t s ,

O b s e r v e t h a t we d i d n ' t u s e a s s o c i a t i v i t y o f only a l e f t i d e n t i t y .

0

;

1 c o u l d have been

9 . 1 1 4 C O R O L L A R Y ( C l a y - M a l o n e ( 1 ) ) . A nr. w i t h i d e n t i t y on a f i n i t e s i m p l e rjroup r i s a f i n i t e p r i m e f i e l d . Proof.

r c a n n o t h a v e a c o m p o s i t e o r d e r by

9 . 1 1 3 . Hence r i s a simple p-qroup, t h u s c o i n c i d i n a with i t ' s a n d we c a n a p p l y c e n t e r . So r i s isomorphic t o H P 9.109.

9 . 1 1 5 COROLLARIES (Krimmel ( 2 ) ,

Clay-Halone

( l ) , Clay-Doi ( l ) ,

Ligh ( 9 ) ) . The f o l l o w i n g g r o u p s T c a n n o t be t h e a d d i t i v e a r o r r p s o f near-rings w i t h identity: ( a ) groups o f composite o r d e r i n wh;ch t h e l a t t i c e of n o r m a l s u b q r o u p s i s l i n e a r l y o r d e r e d (e.cj. S n ( n > 3 ) ) , ( b ) simple groups of composite o r d e r (e.9.

An (nr4)),

( c ) f i n i t e non-abel i a n o r o u p s w i t h e x a c t l y one p r o p e r normal non-zero s u b q r o u p , ( d ) n o n - c y c l i c clroups o f s q u a r e - f r e e o r d e r .

P r o o f . E v i d e n t l y 9 . 1 1 4 =z ( a ) =z ( b ) a n d 9 . 1 1 2 -> ( d ) . I n ( c ) , r must be o f c o m p o s i t e o r d e r s i n c e o t h e r w i s e r i s a n o n - a b e l i a n p-qroup, hence o f o r d e r pk w i t h kr3. I n t h i s c a s e , r h a s a t l e a s t two n o n - t r i v i a l n o r m a l s u b g r o u p s ( s e e e . g . ( R o t m a n ) , C o r . 5.5 a n d Ex. 5.2). We now m e n t i o n w i t h o u t p r o o f some more r e s u l t s o n t h i s s u b j e c t .

If N i s f i n i t e such t h a t t h e i n v a r i a n t subgroups o f ( N , + ) f o r m a c h a i n t h e n 1.1 i s i s o m o r p h i c t o a r i n g Z Pn*

$9 MORE CLASSES OF NEAR-RINGS

330

9.116 THEOREM

( a ; ( L i g h ( 9 ) ) . T h e r e i s no n r . w i t h i d e n t i t y d e f i n a b l e on a t o r s i o n d i v i s i b l e g r o u p . ( b ) ( C l a y - D o i ( 1 ) ) . The same h o l d s f o r and

Am: =

u An. n c IN

Sm:

=

U IN 'n

nE

(1)). There a r e a l s o no n r . ' s d e f i n a b l e on q e n e r a l i z e d q u a t e r n i o n F r o u p s .

( c ) (Clay-Maxson

( d ) ( L i g h ( 1 3 ) ) . T h e r e do e x i s t n r . ' s w i t h i d e n t i t y on p e r f e c t groups ( t h a t a r e qroups coincidincl w i t h i t s commutator s u b g r o u p ) ( c f . Liqh ( 9 ) ) .

( e ) See Johnson ( 4 ) f o r the n r . ' s on the d i h e d r a l nroups D p n o f o r d e r 2 n . T h e r e a r e n o n r . ' s EV, on D Z n f o r odd t i ( t h i s f o l l o w s f r o m 9.111), f o r t h e o n l y o n e s e x i s t on D 4 p ( P E P) . They a r e z e r o - s y m m e t r i c a n d normal N - s u b g r o u p s a n d l e f t i d e a l s c o i n c i d e ( a n d a l l l e f t ideals are annihilator l e f t ideals). There a r e ( u p t o isomorphism) 7 n r . ' s w i t h i d e n t i t y on D 8 ( p . 4 1 8 ) and ( a q a i n u p t o isomorphism) j u s t one o n D ( p ~ I l ' \ { Z l ) . T h e r e i s j u s t one s u c h 4P n r . on t h e i n f i n i t e d i h e d r a l g r o u p ( L o c k h a r t (1),(3)). ( f ) (Clay-Maxson ( 1 ) ) . A l l n r . ' s w i t h i d e n t i t y d e f i n a b l e on p-groups w i t h e x a c t l y one s u b c r o u p o f o r d e r p a r e commuta t i ve r i n o s . ( T h i s f o l l o w s from 9.108 and ( c ) s i n c e a group a s d e s c r i b e d above i s e i t h e r c y c l i c o r a o e n e r a l i z e d q u a t e r n i o n group.)

4 . ) N E A R - R I N G S W I T H O T H E R P P q P E R T I E S OFJ G I V E N G R O U P S

Now we b r i e f l y s t u d y n r . ' s w i t h s p e c i a l p r o p e r t i e s ( o t h e r t h a n h a v i n r l an i d e n t i t y ) c n some q r o u p ( r , t ) . We w i l l o n l y c i t e t h e r e s u l t s o r e v e n o n l v t b e n a o e r s w h i c b a r e concerned w i t h t h e s e t o p i c s . See a l s o t h e c h a p t e r s concerning t h e types o F n e 3 r - r i n g s i n dis::ission. For ex?nple,

33 1

9d Near-rings on given groups

t h e r e a r e no n e a r - f i e l d s d e f i n a b l e on n o n - a b e l i a n n r o u p s ( 8 . 1 1 ) , a n d so o n .

Ide s t a r t w i t h n r . ' s w i t h c h a i n c o n d i t i o n s . G e n e r a l i z i n n 9 . 1 0 2 one q e t s

9 . 1 1 7 THEqREFI ( L i n h ( 3 ) ) . L e t N b e a n r . w i t h DCC on m o n o n e n i c N - s u b n r o u p s o n t h e s i m p l e nrOlJp ( P i , + ) such t h a t N d $. { O I . Then t i i s e i t h e r t h e z e r o - n r . o r a f i e l d . 9.118 R E M A R K For a d e t a i l e d s t u d v o f n r . ' s N o n a nroup which f u l f i l l t h e D C C o n m o n o n c n i c M - s u b n r o u p s a n d t h e " A C C on p r i n c i p a l a n n i h i l a t o r l e f t i d e a l s " ( i . e . each ( 0 : x ) c c ( 0 : x 2 ) s ( o : x 3 )C terminates) s e e Lioh-RarakotaiahReddy ( 1 ) .

...

9 . 1 1 9 THEQPEE1 (Timn ( 3 ) ) . (T',+) i s t h e a d d i t i v e nroup of a ( n o t n e c e s s a r i l y a s s o c i a t i v e ( ! ) ) near-rirrrr i n which every non-zero element h a s a r i o h t i n v e r s e i f f r i s invariantly simple a n d e v e r y y t r h a s ( t h e s a m e ) p r i m e o r d e r . The q u e s t i o n c o n c e r n i n n t h e a d d i t i v e c r o u p o f n e a r - f i e l d s i s s e t t l e d by t h e f o l l o w i n n t h e o r e m .

9 . 1 2 0 T H E O P E V (Timm ( 3 ) ) . The f o l l o w i n g c o n d i t i o n s o n (r,+) are equivalent:

a vroup

r r

i s t h e a d d i t i v e nroup of a n e a r - f i e l d . i s a b e l i a n and t h e a d d i t i v e nroup of a n r . w i t h r i o h t c a n c e l l a t i o n law. ( c ) 'I i s t h e a d d i t i v e n r o u p o f a v e c t o r s p a c e o v e r some field. ( d ) r i s t h e a d d i t i v e nroup of a conmutative f i e l d . ( e ) r i s the a d d i t i v e nroup o f an a l t e r n a t i v e f i e l d . ( f ) T h e r e i s Some P E P s u c h t h a t r i s t h e d i r e c t sum o f t h e nroups ( Z P , t ) o r r i s a d i r e c t sum o c c o p i e s o f

(a) (b)

(@,+). ( g ) r i s a b e l i a n a n d e i t h e r e a c h e l e m e n t h a s t h e same prim.; 91.d.r o r r i s ~ s r : i o n ~ r - e ed i v j s i b l e .

332


F i n a l l y , we c o n s i d e r t h e a d d i t i v e g r o u p o f d g n r . ' s a n d o f i n t e gral n r . ' s . 9.121 REMARKS ( a ) (Ligh ( 1 0 ) ) . There a r e just 3 non-isomorphic d g n r . ' s

o n S6 and a t l e a s t 3 on S n ( n

2 5 ,

n j 6).

There a r e p r e c i s e l y 3 non-isomorphic d g n r . ' s d e f i n a b l e ( p E IP) , b u t none on t h e i n f i n i t e d i h e d r a l g r o u p on D 2P Dm ( L o c k h a r t ( 1 ) , ( 3 ) ) . ( b ) ( L i g h ( 1 3 ) ) . The a d d i t i v e g r o u p o f a s i m p l e d g n r .

is

perfect. ( c ) D g n r . ' s w i t h i d e n t i t y on f r e e g r o u p s a r e e x t e n s i v e l y s t u d i e d i n Zeamer ( 2 ) . ( d ) D g n r . ' ~o n g r o u p s

group

r'

r,

i n which t h e index of t h e d e r i v e d

i s p r i m e , a r e c o n s i d e r e d i n Chandy ( 3 ) .

( e ) ( M a l o n e ( 7 ) ) . T h e r e a r e e x a c t l y 16 d g n r . ' s on a g e n e r a l i z e d q u a t e r n i o n g r o u p . A l l o f them a r e d i s t r i b u t i v e . ( f ) More on d i s t r i b u t i v e n e a r - r i n g s

on g i v e n g r o u p s c a n be

found i n Jones ( 1 ) and W i l l h i t e ( 1 ) . ( 9 ) From 9 . 5 1 we k n o w t h a t i n t e g r a l n e a r - r i n g s groups

r

force

r

t o be n i l p o t e n t . I f

r

on f i n i t e

i s non-abelian

o f o r d e r p 3 ( P E I P ) w i t h p 3 = 2 ' t l t h e n t h e r e a r e no i n t e g r a l n r . ' s N d e f i n a b l e on r', s u c h t h a t N h a s a t l e a s t o n e r i g h t i d e n t i t y -1 0 . I f Irl = p t l , P E I P , p f 2 t h e n e i t h e r a g a i n no s u c h N e x i s t s o r p t l i s a power p t l = 2" o f 2 a n d N i s a G a l o i s - f i e l d ( O l i v i e r (2), H e a t h e r l y - O l i v i e r ( 2 ) ) . (h)

(Lawver ( 3 ) ) . A l l n e a r - r i n g s o n Zm a r e p l a n a r . There P a r e no i n t e g r a l p l a n a r n r . ' s d e f i n a b l e o n Z y , b u t t h e r e a r e some on Z F ( w i t h c h a r a c t e r i s t i c f 5 ! ) .

( i ) "H-monogenic" n e a r - r i n g s ( s e e 9 . 2 7 5 ) a r e g e n e r a l i z a t i o n s o f i n t e g r a l n e a r - r i n g s . A d d i t i v e g r o u p s o f H-monogenic near-rings a r e studied in O l i v i e r ( 2 ) a n d HeatherlyOlivier (3).

9e

Ordered near-rings

333

I f a g r o u p r i s g i v e n by a p r e s e n t a t i o n , i t i s a h i g h l y nont r i v i a l m a t t e r t o c h a r a c t e r i z e a l l n e a r - r i n g s on r . F i r s t studies i n t h i s directions (including "pre-near-rings" ( = m u l t i p l i c a t i v e l y n o n - a s s o c i a t i v e n e a r - r i n g s ) ) can b e found i n L o c k h a r t ( 1 ) and L a x t o n - L o c k h a r t ( 1 ) .

9.122 DEFINITION A nr. N i s c a l l e d p a r t i a l l y ( f u l l y ) ordered by 5 i f

(a) (b)

I makes

i n t o a p a r t i a l l y ( f u l l y ) ordered nroilo.

(N,t)

n,n'EN:

(nrO

A

n'rO

=>

nn'rO).

" O r d e r e d " means " p a r t i a l l y o r d e r e d " .

9.123 REVARKS

( a ) T h u s an o r d e r e d n e a r - r i n n i s a n r . w h e r e (PI, ? nn'2O.

and

( b ) The s t a n d a r d w o r k on o r d e r e d a l a e b r a i c s y s t e m s ( s e n i -

g r o u p s , clroups, r i n n s a n d f i e l d s ) i s ( F u c h s ) .

( c ) F o r a n o r d e r e d n e a r - r i n g we w i l l w r i t e simply

(N,+,*,s)

or

(3,s).

( d ) P a r t s of o u r d i s c u s s i o n i s i m p l i c i t in ( G a b o v i c h ) .

O f course,

nl

S

n 2 and 0 s n i m p l i e s

nlnS

n 2 n i n an o r d e r e d n r . N .

Some a u t h o r s ( K . B . P . Rao ( l ) , f o r i n s t a n c e ) r e q u i r e t h a t a l s o n n l s n n 2 follows. Cf. 9 . 1 5 2 ( b ) , ( d ) .


334

9 . 1 2 4 NOTATION I4e a d o p t t h e u s u a l c o n v e n t i o n s t o w r i t e n < n ' , n ? n ' , n > n ' , nl1 n ' ( n a n d n ' a r e incomparable, i . e . neither nrn' nor n ' s n holds). " P a r t i a l l y o r d e r e d " w i l l be a b b r e v i a t e d by " P . o . " , " f u l l y o r d e r e d " by " f . o . " . J u s t a s i n t h e t h e o r y o f o r d e r e d n r o u p s o r r i n n s , i t i s more c o n v e n i e n t t o work w i t h t h e s e t o f " p o s i t i v e " e l e m e n t s i n s t e a d of the o r d e r r e l a t i o n i t s e l f : 9 . 1 2 5 THEOREkt ( a ) Let ( E l , + , * , < ) be a p . 0 . n r . ; t h e n t h e "-___-positive cone" P : = P: = { n E l ! l n r o l fulfills 5

(a) PtP P. ( R ) P n ( - " ) = (01, where, as u s u a l , ( y ) d nEpI: n t P = P t n . ( 6 ) P * P c P.

-P:

= {nlnir)).

( b ) C o n v e r s e l y , f o r e v e r y s u b s e t P o f a n r . rl f u l f i l l i n i

( a ) - ( 6 ) we clet a n o r d e r e d n r . n sP n ' : n ' - n ~ P .

(N,sp)

via

( c ) T h i s c o r r e s p o n d e n c e between o r d e r r e l a t i o n s a n d subsets w i t h ( a )

-

(6) is

t h a t means t h a t

1-1,

The p r o o f o f 9 . 1 2 5 i s e a s y a n d l e f t t o t h e r e a d e r . 9 . 1 2 5 e n a b l e s u s t o s a y t h a t " t h e n r . PI i s o r d e r e d by P " . The f o l l o w i n c l r e s u l t i s o b v i o u s . 9 . 1 2 6 PROPOSITI3PI Let FI be o r d e r e d by P.

( a ) sP (b)

i s a f u l l o r d e r

sP i s t r i v i a l ( i . e .

P w(-P)

n I, n ' < - >

=

N. n = n ' )

There i s no p l a c e f o r f i n i t e near-rirlns i n t h i s s e c t i o n :

P

=

{Ol.

335

9e Ordered near-rings

9 . 1 2 7 PROPOSITION Every n o n - t r i v i a l l y o r d e r e d n e a r - r i n o i s infinite. Proof.

3

nErl:

n>O.

B u t then

... .

n < n t n = 2n r ) . b - a ~ U - > b - a = b , w h i c h i s a c o n t r a diction, too. Hence U = fl and N i s c y c l i c a n d i n f i n i t e ( 9 . 1 2 7 ) . The map h : N = A + Z i s a n o r d e r - i s o m o r p h i s m z za between t h e a d d i t i v e groii?s.

-.

340

0 9 MORE CLASSES OF NEAR-RINGS

By 9.137, (usual

= -z*z'.

z*z':

z

+

2 a n d Z'

I n an o r d e r e d n r .

N one can ask, I n oeneral

Rrx]

are order-isomorphic

PI =o H %

Hence i n any c a s e

-2.

m i q h t be r e l a t e d . but for

H = (H,+,*) h': = (Z,+,*)w i t h

N i s isomorphic t o t h e r i n n

multiplication) or

how

Inn'\

via

.

and

In1 I n ' '

t h e r e i s no d i r e c t r e l a t i o n s h i p ,

o f 9.142(b) we n e t t h e f o l l o w i n n r e s u l t w h i c h

we s t a t e w i t h o u t p r o o f .

9.145

T H E O R EY (Pi12 (4)).

In

we h a v e f o r a l l

(R[x],t,o)

P,qER[x]: (a)

l p o q l = / p l o j q l c->

( p c o n t a i n s o n l y e ~ c no r

(420) v

o n l y odd d e q r e e s ) .

(b)

l p o q l 5 l p l o l q l

(qcr)) A ( t h e c o e f f i c i e n t s o f t h e o r e a t e s t e v e n and n r e a t e s t odd d e n r e e o f p h a v e t h e same s i n n ) .

(c)

l p o q / 2 I p l o ' q j

( q < O ) A ( t h e c o e f f i c i e n t s of t h e q r e a t e s t even and o r e a t e s t odd p h a v e o p p o s i t e sinn).

deqree of

J . Zemmer h a s s h o w n t h a t a d i r e c t s u m o f

f.0.

r i n n s can be

f . o r d e r e d i f f a l l b u t a t m o s t o n e o f t h e summands a r e z e r o r j n n s . Me now o b t a i n a s i m i l a r r e s u l t f o r n r . ' s ments on t h e s t r u c t u r e o f

9.146 T H E O R E M l f one of

the

i m p l y i n n some s t a t e -

nr's.

f.0.

S

N =

@

i s

Ni i-1

Nils

then i n a l l but a t most

f.0.

a l l p o s i t i v e elements ( i n the order

i n d u c e d b y N v i a t h e p r o j e c t i o n R a p s ) a n n i h i l a t e Fli from the r i q h t . Proof.

3

Assume t h a t

3

i,jE{l,

-

3

..., sl,

i I- j 101 A N . n ~j a n d ns;cNj,

O n -> ==> n n ' > O -R[x! of 9.142(b) i s s t r i c t l y ordered i f R i s i n t e q r a l ) o r i f PI c o n t a i n s a l e f t i d e n t i t v t h e n s = 1 .

P r o o f . F o r s t r i c t o r d e r s t h i s i s i m m e d i a t e from 9 . 1 4 6 . S

1 ei i=l eiE?i As i n 3 . 4 3 , e i i s a l e f t identity i' a n d a q a i n w e can employ 9 . 1 4 6 t o n e t s = 1 .

I f ri c o n t a i n s a l e f t i d e n t i t y e , l e t with i n :Ii

9.148 C r l R O L L A R V Every 1 - s e m i s i m p l e is simple.

f.0.

e =

!ienl

nr.

with DCCL

Proof: by 9.137, 5.31 a n d 9.147. 9 . 1 4 9 REM4RK rlne c a n n o t i m n r o v e 9 . 1 4 6 t o n e t t h e e x a c t a n a l o n u e o f Zemner's r e s u l t : t a k e f o r N l , F 1 2 any f . 0 . c o n s t a n t non-zero near-rinns a n d use the lexicoaraphic order. Exanininn a b s t r a c t affirle near-rinas oives a stranoe r e s u l t w h i c h shows t h a t " n e a r l y n o " a . a . n . r . c a n b e f u l l y o r d e r e d : L e t PI be a n a . a . n . r . s u c h t h a t o r d e r e d . Then I! c a n be f . 0 . FI o H c = v No i s a z e r o r i n c l ) .

rJo,Y,

9 . 1 5 0 TIIE3PE'f -___-

(Q)

A

are fully ( l l c = {'I)v

P r o o f . =>: ( a ) F i r s t we show t h a t ( N , + ) = ( N C , + ) G ( N , , + ) ( 9 . 7 3 ( a ) ) must have t h e l e x i c o o r a p h i c o r d e r . I f n r O t h e n n c = n9lc); l i k e w i s e n s ' l i f n p l f e s

n C SO. I f nc = 0

then

norO),

n

n 0 >O.

n E N : n20 9) v (nc i . e . the lexicographic order.

So He q e t f o r A

n > 0

r) A

342

S 9 MORE CLASSES OF NEAR-RINGS

( b ) Assume n o w t h a t N O H c {Ol. q i n c e N o = N d , we c a n f i n d n o E N 0 and PtccN, w i t h n c > 9 and noncr) bv ( a ) . B u t nn, = nonc O nonAn. L e t n : = no>O and n ' : = n j t n c > O . Then n n ' = n o n ; t t n o n c = nonh n n " ) .

n i s p o s i t i v e d e f i n i t e : V n'EN : n n ' r 0 .

See P i l z ( 1 ) f o r r e s u l t s on t h e s e concepts.

( e ) ( P i l z ( 8 ) ) . L e t N be a n r . w i t h ( N , t ) = ( N o , t ) ~ ( ! 4 c , t ) ( c f , 1 . 1 3 ) , where N o a n d Pic a r e f . 0 . n . r . ' s (by P o + . T h e n t h e f . 0 . of No a n d N c can be e x t e n d e d t o a f . 0 . on N i f f po~:Po 1 pC€PC nocNo: p o ~ ( n 0 o + p c ) ~ P o . In t h i s c a s e t h e o r d e r i s the " l e x i c o o r a p h i c " one d e t e r m i n e d by n o t n C -z 0 ( n C > O ) v ( n C = r) A n o 2 0 ) (see 9.150). ( f ) I t i s hard t o q e t f u l l orders in "non-deaenerated"

n e a r - r i n o s ( 9 . 1 4 1 ) . B u t i t i s very n a t u r a l t o l o o k f o r l a t t i c e - o r d e r s ( i . e . such t h a t (N,s) i s a l a t t i c e ) . F o r i n s t a n c e , M(T), where 7 i s a f . 0 . q r o u p , c a n be a i v e n a l a t t i c e o r d e r by

msm' :

y

~ :r m ( y ) m l ( y ) .

F o r d e t a i l s a n d c o n n e c t i o n s t o " F - n e a r - r i nns" N ( t h e s e a r e s u b n r . ' ~a n d s u b l a t t i c e s o f a d j r e c t IT l i i o f f . 0 . n r . ' s €cllc, l a t t i c e - o r d e r e d product if1

( . . . , n i ,... ) I ( . . . , n ; ,... ) : c-> i c I : n i s n !1) and t o v e c t o r - n e a r - r i n q s ( F - n r . ' s , where is 3 s u b d i r e c t p r o d u c t o f t h e N i l s ) s e e O i l z (I), B b a n d a r i R a d h a k r i s h n a (I) a n d P a d h a k r i s h n a ( 1 ) . by

(9)

Kerby ( 1 ) , ( 3 ) , ( 5 ) a n d G r o g e r ( l ) ,( 2 ) s t u d i e d o r d e r e d n e a r - f i e l d s . A n f . F i s f o r m a l l v r e a l i f -1 i s n o t

344

89 M O R E CLASSES OF N E A R - R I N G S t h e sum of p r o d u c t s of s q u a r e s . F can be f u l l y o r d e r e d i f f F i s formally real (Groger ( 1 ) ) . ( h ) Extensions o f p a r t i a l orders t o f u l l orders a r e studied i n K . B . P . Rao ( 1 ) . (i) N a t a r a j a n ( 3 ) and K . B . P .

o r d e r e d N-groups. ( j ) See a l s o Kiisel

(1).

Rao ( l ) , ( 2 ) a l s o c o n s i d e r e d

345

9f Regular near-rings

f ) REGULAR N E A R - R I N G S Von Neumann r e g u l a r r i n g s p l a y a n i m p o r t a n t r 8 1 e i n r i n g t h e o r y . T h e y g e n e r a l i z e some p r o p e r t i e s o f n e a r - f i e l d s t o a m u c h w i d e r class of rings.

This concept n o t only transfers t o near-rings,

i t i s a l s o m o t i v a t e d by t h e f a c t ,

types o f near-rings

t h a t some o f t h e m o s t i m p o r t a n t

are r e g u l a r (see 9.154).

9.153 DEFINITION A n e a r - r i n g i s c a l l e d r e g u l a r i f Vn E N 9.154

3x

E

N:nxn

=

n

EXAMPLES R e g u l a r n . r . I s a r e o b v i o u s l y : ( a ) M ( 7 ) and M o ( r )

(Beidleman ( 1 0 ) ) .

( b ) Constant near-rings.

( c ) D i r e c t sums a n d p r o d u c t s o f n e a r - f i e l d s . (d) Integral planar near-rings

N ( s i n c e f o r n E N we c a n

f i n d X E N w i t h n = xn2 by 8.88 =

(el (N,t,*)

I n 9.153,

(b);

now ( n - n x n ) n

=

n 2 - n 2 = 0 g i v e s t h e r e s u l t (Mason ( 5 ) ) . f o r any group

( N , + ) a n d n*m:

n i f m f O O i f m = O

xn can be c o n s i d e r e d as a " p r i v a t e r i g h t i d e n t i t y "

and nx as a " p r i v a t e l e f t i d e n t i t y " f o r n.

n has

I f N E ~ , and

an i n v e r s e x t h e n n x n = n , o f c o u r s e .

9.155 REMARKS ( a ) I n 9.153,

n x and xn a r e i d e m p o t e n t .

( b ) By 9 . 1 5 4 ,

regular near-rings are not necessarily

abel ian. ( c ) Homomorphic images,

d i r e c t sums a n d d i r e c t p r o d u c t s

o f regular near-rings are regular.

7.33,

1.86 and 1.88,

By 9 . 1 5 4

every (zero-symmetric)

r i n g c a n b e embedded i n a ( z e r o - s y m m e t r i c ) regular near-ring.

nearsimple

9 . 1 5 4 ( a ) a l s o shows t h a t i n

general a regular nr.

(P,)-(P4)

(a),

has n e i t h e r t h e IFP n o r

(see 9.1 and 9 . 4 )

09 M O R E CLASSES OF N E A R - R I N G S

346

( d ) By 9 . 1 5 4 ( a ) , s u b n e a r - r i n g s o f r e g u l a r n r . ' s a r e n o t regular in general. N e v e r t h e l e s s , s e v e r a l c o n n e c t i o n s t o I F P - n r . ' s and t h e i r p r o p e r t i e s w i l l s h o w u p . We now c h a r a c t e r i z e r e g u l a r n e a r - r i n g s a n d d i s p l a y some o f t h e i r p r o p e r t i e s a f t e r w a r d s . 9.156 T H E O R E M (Beidleman ( l o ) , Ligh ( 7 ) ) . Let N < = > v n E N 3 e = e 2 E N : Nn = Ne.

EN,.

N i s regular

P r o o f . =+ : T a k e X F N w i t h n x n = n . T h e n Nn = N ( x n ) d o e s t h e j o b by 9 . 1 5 5 ( a ) . + = : T a k e n c N . Then N n = Ne

Ne, t h e r e i s some x E N w i t h x n = e . S i n c e N ' f l , , n E Nn = Ne, h e n c e n = y e f o r some y t N , a n d we g e t n = y e = y e e = = yexn = n x n .

f o r some i d e m p o t e n t e .

Since e

F

9.157 COROLLARY (Beidleman ( 1 0 ) ) . A r e g u l a r n e a r - r i n g with i d e n t i t y c o n t a i n s no n o n - z e r o n i l N - s u b g r o u p s . H e n c e we m i g h t l o o k a t r e g u l a r n e a r - r i n g s w i t h o u t n i l p o t e n t elements. 9 . 1 5 8 T H E O R E M ( L i g h ( 7 1 , Chao ( I ) , O s w a l d ( 9 ) ) . L e t N {Ol be a r e g u l a r n e a r - r i n g w i t h i d e n t i t y . E q u i v a l e n t a r e :

(a) N

h a s no n o n - z e r o n i l p o t e n t e l e m e n t s . ( b ) A l l i d e m p o t e n t s of N a r e c e n t r a l . ( c ) N i s a s u b d i r e c t product o f n e a r - f i e l d s . P ro o f . ( a ) * ( b ) h o l d s by 9 . 4 3 ( 6 ) . ( b ) + ( c ) : By 1 . 6 2 , N i s isomorphic t o a s u b d i r e c t product of s u b d i r e c t l y i r r e d u c i b l e n r . ' s . N i ( i E I ) . These Nils a r e r e g u l a r by 9 . 1 5 5 ( c ) , EY, a n d f u l f i l l t h e c o n d i t i o n ( b ) , = No

t o o . Let A : = n ( O : e ) , w h e r e e r u n s o v e r a l l i d e m p o t e n t s f 0 and 1 in N i . Since each e i s c e n t r a l ( 0 : e ) (and hence A ) a r e i d e a l s . I f ( 0 : e ) = I01 t h e n e = 1 , a c o n t r a d i c t i o n . By 1 . 6 0 , A f {0}

+

347


T a k e a E A , a f 0 . Now a x a = a f o r some x c N .1 . I f e 2 = e E N. t h e n a e = 0 , h e n c e e a = 0 , w h e n c e 1 e c ( 0 : a ) . ax i s idempotent by 9.155 ( a ) . I f ( 0 : a x ) = {O}, a x = 1 a n d e = e ( a x ) = ( e a ) x = Ox

=

If (0:ax) { a } , we g e t a E ( O : a x ) , hence a = ( a x l a = a ( a x ) = 0 , again a c o n t r a d i c t i o n . T h i s shows t h a t 0 and 1 a r e t h e o n l y 0 and n = n x n idempotents in N i . If n c N i i s =

0, a contradiction.

+

t h e n nx a n d x n a r e 0 and hence = 1 (by 9.155 ( a ) ) . Hence N i i s a n e a r - f i e l d . ( c ) * ( a ) i s t r i v i a l . The e q u i v a l e n c e ( b ) o ( c ) i s t r u e w i t h o u t t h e a s s u m p t i o n M E ? " , . T h i s r e s u l t ( and i t s proof

) show

9 . 1 5 9 COROLLARIES ( B e i d l e m a n ( l o ) , L i g h

( 7 ) , Heatherly ( 8 ) ,

Chao ( I ) , M a r i n ( I ) ) . ( a ) A r e g u l a r n e a r - r i n g whose i d e m p o t e n t s a r e c e n t r a l i s a b e l i a n , 2 - s e m i s i m p l e a n d an I F P - n e a r - r i n g ( 9 . 3 7 ) . ( b ) A r e g u l a r d g n r . whose i d e m p o t e n t s a r e c e n t r a l i s a semisimple r i n g . ( c ) A regular near-rinq N with i d e n t i t y 1 4 0 i s a nearf i e l d i f f 0 a n d 1 a r e t h e o n l y i d e m p o t e n t s i n N. ( d ) A r e g u l a r nr. w i t h D C C I whose i d e m p o t e n t s a r e c e n t r a l

is a finite direct

sun1

of n e a r - f i e l d s .

( e ) I n a r e g u l a r nr. w i t h i d e n t i t y whose i d e m p o t e n t s a r e c e n t r a l , e v e r y N-subgroup i s a l e f t i d e a l . ( f ) A r e g u l a r nr. w i t h i d e n t i t y i s i n t e g r a l i f f i t i s a near-f iel d. This gives another c h a r a c t e r i z a t i o n of r e g u l a r n e a r - r i n g s 9 . 1 6 0 T H E O R E M (Chao ( I ) ) Suppose N

= PIo nilpotent elements. N i s regular-

h a s no n o n - z e r o Na i s a d i r e c t summand

of N f o r each a t N . P r o o f . =, : By 9 . 1 5 9 ( a ) a n d ( e ) , N i s a b e l i a n a n d e a c h -__ Na s R N . B u t Na = Ne f o r some i d e m p o t e n t e , w h e n c e ( N , + ) = Ne i ( 0 : e ) by 1 . 3 3 .

09

348 e :

MORE CLASSES OF NEAR-RINGS

l e t L b e a l e f t i d e a l of N w i t h ( N , + )

For n E N ,

=

NnTL.

n

=

n.1

There a r e m E N a n d 1 E L w i t h 1 =

n m n + n l by 2 . 2 9 .

B u t nl

=

=

m n + l , whence

=

-nmn+n

E

NnnL

=

{O)

b v 1 . 3 4 . So n = n m n .

In Ligh - Utumi elements then N We a l s o mention with I N \ > 1 . F o r

( 1 ) i t i s shown t h a t i f N E ? ? ~ has i s r e g u l a r i f f nN = nNn holds f o r a n o t h e r r e s u l t o f Ligh ( 2 ) : L e t N e a c h n E N t h e r e i s e x a c t l y one x E

no nilpotent a l l n r N. be a d g n r . N with n x n = n

iff N is a near-field. R e g u l a r n r . ’ s w i t h one ( a n d hence a l l ) of t h e t h r e e c o n d i t i o n s s t u d i e d i n 9 . 1 5 8 a r e o b v i o u s l y of p a r t i c u l a r i m p o r t a n c e . They deserve a s p e c i a l n o t a t i o n . 9.161 DEFINITION A regular near-ring N i s called strongly N EN, a n d i f N f u l f i l l s t h e c o n d i t i o n s regular if { O } in 9.158.

+

9 . 1 6 2 T H E O R E M (Marin ( I ) ) . N ~ f l ~ ~i sm s ,t r o n g l y r e g u l a r i f f V n E N

~ x E N :n

P-r o o f . +: Take -

n

E

2 xn . N. Then n =

=

n x n f o r some x

E

N.

Hence

x n i s i d e m p o t e n t , hence c e n t r a l .

.

2 n = n x n = xnn = xn = : Let n E N. Then n = x n 2 f o r some x r N . Hence n 2 = x n 3 , a n d s o o n . Thus t h e r e c a n n o t be some k E N k-I f 0 , and N i s shown t o b e a with n k = 0 , b u t n n r . w i t h o u t n i l p o t e n t e l e m e n t s . Now n 2 = n x n 2 , whence ( n - n x n ) n = 0 , hence n ( n - n x n ) = 0 . So

1/Je g e t ( n - n x n ) ’ consequently n =

=

n(n-nxn) nxn.

- nxn(n-nxn)

=

0-0

=

0,

349


We r e m a r k t h a t M ( T ) a n d M o ( r ) f o r m e x a m p l e s of r e g u l a r , b u t n o t strongly regular near-rings. Integral p l a n a r near-rings are e x a m p l e s o f s t r o n g l y r e g u l a r n e a r - r i n g s . Many r e s u l t s on s t r o n g l y r e g u l a r n e a r - r i n g s c a n be f o u n d i n Mason (5). We p r e s e n t some o f t h e s e r e s u l t s : 9 . 1 6 3 T H E O R E M L e t N be s t r o n g l y r e g u l a r . ( a ) E v e r y p r i m e i d e a l of N i s maximal ( c f . 2 . 7 2 ) . (b)

\In

c N

3

X E

N : n = xn2

A

x is invertible.

( c ) Every N-subgroup o f N i s a ( t w o - s i d e d ) i d e a l . 2 ( d ) Every i d e a l I of N f u l f i l l s I = I . P r o o f . ( a ) L e t P be a p r i m e i d e a l a n d s u p p o s e t h a t P c M c N , M S N . I f m E M \ P t h e r e i s some X E M w i t h n

0 = m - x m L = ( I - x m ) m . By 2 . 6 1 a n d t h e I F P we g e t e i t h e r m E P ( a c o n t r a d i c t i o n ) o r 1 - x m E P , whence 1 E M , again a contradiction. ( b ) n = a n 2 f o r some a E N and a = z a 2 f o r some z . L e t x : = I - z a + a . Then x n 2 = ( I - z a + a ) n 2 = n 2 - z a n 2 + a n 2 = 2 = n2-za(an ) n + n = n2-n2+n = n , a n d xa = ( I - z a + a ) a = 2 = a 2 . I f x i s c o n t a i n e d i n a maximal i d e a l = a-a+a M, a 2 E M , b y 2 . 7 2 hence a E M . SO 1 E M , a c o n t r a d i c t i o n . Hence x i s a u n i t . ( c ) Lde k n o w a l r e a d y ( 9 . 1 5 9 ( e ) ) t h a t e v e r y N-subgroup S of N i s a l e f t i d e a l . I f s E S and n E N then s

E

Ns

Ne f o r some i d e m p o t e n t

=

e , hence s

=

ne.

N o w e i s c e n t r a l . Hence s n = Ken = n n e E Ne = N s s S . 2 ( d ) Of c o u r s e , I c I . I f i E I t h e n t h e r e i s some X E

N with i

=

xi2

=

(xi)i

E

I

2

.

I n f o r m a t i o n c o n c e r n i n g t h e r a d i c a l s o f a r e g u l a r n r . was o b t a i n e d i n J o h n s o n ( 6 1 , which we s t a t e w i t h o u t p r o o f . 9.164

T H E O R E M Let N

( a ) %,,2(~)

=

be r e g u l a r . Then {OI

89 MORE CLASSES OF N E A R - R I N G S

350

#

( b ) Every minimal l e f t i d e a l

{O}

i n N i s a minimal

N - s u b g r o u p. ( c ) I f N has t h e DCCN t h e n N i s r e g u l a r i f f N i s 2-semis i m p l e w i t h n EN^ f o r a l l n E N . ( d ) I f N has t h e DCCN t h e n maximal i d e a l s c o i n c i d e with orirnitive ideals. S t i l l more

i n f o r m a t i o n c a n be found

( 1 ) and Ramakotaiah ( 3 ) .

i n Choudari-Goyal

We s h a l l c o n s i d e r r e g u l a r n e a r - r i n g s

o f t h e t y p e M (1') i n 9 9 ( h ) . S

a ) TAME NEAR-RINGS

I n t h i s c h a p t e r we i n v e s t i g a t e a c l a s s o f n r . ' s

which i s c l o s e l y

r e l a t e d t o c o m p a t i b l e n e a r - r i n g s as d e f i n e d i n 7 . 1 3 7 . o f t h e r e s u l t s i n t h i s t h e o r y a r e due t o S.D.

D E F I N I T I O N An N - g r o u p

r

i s c a l l e d __ tame i f e v e r y No-

subgroup o f NT i s an i d e a l . a faithful

tame N-group

Hence i n t a m e N - g r o u p s ,

Scott. For the

1.34.

following definition cf. 9.165

Most

r

A n e a r - r i n g N i s ___ tame i f N has

( t h e n N i s c a l l e d t a m e o n Ni'-).

i d e a l s and No-subgroups c o i n c i d e .

There a r e several examples which work f o r d i f f e r e n t reasons. 9 . 1 6 6 EXAMPLES (a) I f N i s 2-primitive (since

,,,r

t h e n N i s tame on

on

has no n o n - t r i v i a l

l e t S ( r ) be the nr. a d d i t i v e l y

(b) I f Inn(r)sSsEnd(r), generated by S.

I f S = I n n ( r ) then

S = E n d ( 1 ' ) t h e n S(T)

=

r,

S(T)

E(1'), and so on.

hence z e r o - s y m m e t r i c and group

Nr

No-subgroups i n t h i s case). =

I(T), if

S ( r ) i s d.g.,

S ( r ) i s tame on t h e S ( T ) -

since every S(r)-subgroup

A i s normal

(because A i s i n v a r i a n t under a l l i n n e r alAtomorphisms), hence an i d e a l by 6.6.

351

9 g T a m e near-rings

( c ) L e t l? be a v a r i e t y o f ( 2 - g r o u p s a n d A EU. Then ( A , + ) 21 i s a tame A and P ( A ) - g r o u p , and P ( A ) i s tame

[XI-

on A .

This holds since B 5

p(b)E B for all

P F

A implies t h a t Po(A) P o ( A ) a n d b E B . Hence a l l f i n i t e

s u m s o f t h e s e e l e m e n t s a r e i n B y w h e n c e B !A by 7 . 1 2 3 . B u t t h e e l e m e n t s o f P ( A ) a r e C o m p a t i b l e by 7 . 1 2 2 ; c c n s e q u e n t l y B i s an i d e a l of p ( A ) A . ( T h e same a r g u m e n t s a r e a p p l i c a b l e f o r A*[x] i K s t e a d o f P ( A ) , w i t h t h e o n l y e x c e p t i o n t h a t Al'lx] a c t s n o t n e c e s s a r i l y i n a f a i t h f u l m a n n e r on A . ( d ) More g e n e r a l l y , e v e r y n e a r - r i n g N b e t w e e n P ( A ) a n d C ( A ) i s tame on A ( c f . a l s o 7.140 and 9 . 1 6 8 ! ) .

( e ) Every ring-module i s tame. Every r i n g w i t h i d e n t i t y i s tame. ( f ) Many m o r e e x a m p l e s w i l l come u p b y 9 . 1 6 8 a n d 7 . 1 3 7 ! S c o t t r e m a r k e d t h a t S(T

i n 9 . 1 6 6 ( b ) i s a l s o ( b y 9 . 1 6 8 we w i l l s a y : " m o r e o v e r " ) c o m p a t b l e on r : I f Y E r a n d i f s E S o r - s c S t h e n e i t h e r s(y+6 - s y = s y + s G - s y = s ( y + i s - y ) o r = s t ~ " o r a l l S t r . N o w : n n ( r c _ S , h e n c e t h e r e i s some n E S(r) w i t h i t r . T h i s e x t e n d s t o a l l f i n i t e is(),+fi) - s y = n 6 f o r a1 s u m s o f e l e m e n t s o f S . H e n c e S ( 1 ' ) i s c o m p a t i b l e on r . T h e s i m i l a r i t y between the c o n c e p t s "tame" and " c o m p a t i b l e " i s r e v e a l e d by

~v~

and NT i s u n i t a r y t h e n 9 . 1 6 7 PROPOSITION ( S c o t t ( 1 7 ) . I f N Nr i s t a m e i f f f o r a l l y , 6 E r a n d n E N o t h e r e i s some m E No w i t h n(y.6) - ny = m 6 . Pr o o f . I f Nr i s u n i t a r y a n d t a m e t h e n e a c h N o & i s an i d e a l o f N T c o n t a i n i n g 6 . Hence n(y.6) - ny E N o & . C o n v e r s e l y , s u p p o s e t h a t n 5 r . I f y E r and ~ E t hA e n y + 6 - y = l ( y + 6 ) - l yN O = my E n ( f o r some Hence A i s n o r m a l . I f y E then n(y.6) - n y = n C ( y + 6 ) - noy a E No).

m E

No).

r, =

6 E A and n E N a 6 E A ( f o r some

89

352


T h i s i s shown by t h e f o l l o w i n g p i c t u r e ( c f . t h e d i a g r a m a f t e r 7.136 !)

tr

9.168

C O R O L L A R Y Every u n i t a r y c o m p a t i b l e N-group i s t a m e .

( 2 0 ) a n d (21), S.D. S c o t t g o e s o n t o t h e s t u d y o f a t y p e o f n e a r - r i n g s between t a m e a n d c o m p a t i b l e n e a r - r i n g s : In ( 1 7 ) ,

9 . 1 6 9 DEFINITION - L e t k b e a c a r d i n a l number. A n N - g r o u p r i s k - t a m e on m~

Nri f

for all n

No w i t h n ( y + S i ) - n ( y )

of a t most k e l e m e n t s S i

=

t

No and

m(si)

Y E

r

t h e r e i s some

f o r any c o l l e c t i o n

i n I'.

H e n c e we g e t f o r u n i t a r y N - g r o u p s : + k - t a m e + ... =. 2 - t a m e + I - t a m e c o m p a t i b l e =r

...

=

tame.

We c i t e some r e s u l t s on 2 - t a m e n e s s w i t h o u t p r o o f .

-9 . 1 7 0 T H E O R E M ( S c o t t ( I ) ,

(81,

(20)). L e t NE%,,,-,~,

be 2-tame

on t h e u n i t a r y N - g r o u p r. ( a ) I f h i s an N-endomorphism o f r then i d - h i s an N - endorno r p h ism , t o o ( b ) I f h i s an N-automorphism of r and i d - h i s a n N a u t o m o r p h i s m , t o o , t h e n - i d i s an N-automorphism.

.

9g Tame near-rings

353

( c ) If A u t N ( T ) c o n t a i n s a f i x e d - p o i n t - f r e e element t h e n - i d i s an N - a u t o m o r p h i s m . ( d ) I f - i d i s an N-endomorphism t h e n ( E n d ( r ) , + , " ) i s a r i n a and E n d ( r ) = E ( T ) . ( e ) I f - i d i s a n N-endomorphism and i f r i s f a i t h f u l w i t h o u t e l e m e n t s o f o r d e r 2 t h e n PI i s a r i n g a n d NI' i s a n N-module. ( f ) If

N r has

r

DCCI a n d A C C I and i f

=

Ali

...; A r

=

where t h e A i l s a n d E ' s a r e indecomposable j i d e a l s of N r t h e n r = s and t h e r e i s a p e r m u t a t i o n p o f { I ,... , n } w i t h A l 1 E p ( , ) ,... , A r = N E p ( r ) . = E,;

...+

E,,

( "Krull -Schmidt-Theorem"). ( 9 ) I f N i s 2 - p r i m i t i v e on I' a s w e l l t h e n N i s a r i n g o r N i s d e n s e i n Mo(I') ( i . e . G

=

(id} i n 4.60).

( h ) I f n o n o n - z e r o homomorphic image o f N i s a r i n a ( N i s then c a l l e d r i n g - f r e e ) and i f N h a s t h e D C C L then N i s f i n i t e .

We n o w m e n t i o n some e l e m e n t a r y f a c t s a b o u t tame n e a r - r i n g s 9.171

PROPOSITION L e t a l l a p p e a r i n g N - g r o u p s be u n i t a r y ( a ) L e t N be tame o n r a n d A a N r . Then N i s tame o n A (b)

and on rlA. I f N i s tame o n T i

r:

=

( i c I ) t h e n N i s tame on

Ori.

i €1 ( c ) I f N i i s tame o n T i

(i

E

I ) then

II N i

i s tame o n

i €1 @Ti.

iEI ( d ) I f N i s 2-semisimple t!ien N i s tame. P r o o f . ( a ) f o l l o w s from 1 . 3 0 , ( b ) i s s t r a i g h t f o r w a r d since A 5 r implies ( . . . 0 , ~ , ~ , . . . ) ( . . . , 6 i , . . . ~ NO

=

,... ) I (. ..

( . . . , 0 , 6 i ,O

E

A , whence A = @ A i

i €1

=

for

A} ( c ) f o l l o w s f r o m t h e f a c t t h a t i f N i i s tame o n r i t h e n N i s tame on I' ( b y ( . . . , n i ,... ) y 1. : = n1 . y1 . ) , t o o .

Ai = { S i

E

Ti

Now a p p l y ( b ) .

,9,.. .)

E

354

89 MORE CLASSES OF NEAR-RINGS ( d ) I f T i ( i E I ) r e p r e s e n t a l l non-N-isomorphic N - g r o u p s o f t y p e 2 t h e n N i s tame o n t h e i r d i r e c t sum ( S . D . S c o t t ) .

A s p l i t t i n g of N d o e s n o t i n d u c e a s p l i t t i n g of

Nr

i n general.

B u t i t d o e s f o r tame n r . ' s . 9 . 1 7 2 THEOREM ( S c o t t ( 1 7 ) . Let N f be tame, u n i t a r y a n d f a i t h f u l .

If N f = A;B

i s t h e d i r e c t sum o f t h e i d e a l s I a n d J t h e n w i t h (0:A) = J a n d ( 0 : B ) = I .

P r o o f . In N = I + J , 1 decomposes as 1 = e l + e 2 , where e 1 ' e 2 a r e c e n t r a l o r t h o g o n a l i d e m p o t e n t s . Let A: = e l l ' a n d B : = e 2 r . N o w J = ( o . A ) , s i n c e JA = = J e l i ' = { o } a n d i f n E ( o : A ) , n = ne + n e 1 2' thep ne:y = 0 h o l d s f o r a l l y F r , b e n c e n e , = 0 a n d n = n e 2 c J. S i m i l a r l y , I = ( o : B ) . Next we show t h a t A 5N I' ( t h e n A g N 1 ' ) . N A = N e l r

=

0

A f o r a l l 6 E A we show t h a t A i s c l o s e d u n d e r a d d i t i o n . L e t e l y l , e l r 2 E A . Then e 2 ( e l y l + e l y 2 ) - e 2 e l y 1 = m e l y 2 f o r some m t N. S i n c e e 2 e l y 1 = 0 we g e t e 2 ( e l ~ l + e l y 2 =) = mely2. M u l t i p l i c a t i o n by e 2 g i v e s e 2 ( e l y , + e , y 2 ) = = e2rnely2 = e e m y 2 = 0 , whence e , y l + e l y 2 E A. 2 1 C o n s e q u e n t l y 5 --N 1', a n d s i m i l a r l y 6 c N r . O b v i o u s l y =

elNrc_elr = A.

r

=

=

ele2y2

Since ( - I ) &

E

0

A+B.

If e l y l =

=

0 . Hence

0

e2y2

r

=

E

A n B then

elyl

=

s

elyl

=

A+B.

W i t h o u t p r o o f we m e n t i o n

n, h a v e

M ( S c o t t ( 1 7 ) . Let N c N O n 9.173 T H E O R El e t N r be u n i t a r y , tame a n d f . g . ;

is f.g.,

the ACCL, and then a n y i d e a l of N T

too.

Next we l o o k , h o w f o r S c - N t h e l e f t i d e a l 'looks l i k e .

tS>

v. g e n e r a t e d

by S

355

9g Tame near-rings

9.174

PROPOSITION ( S c o t t ( 1 7 ) .

L e t N T be tame and M

Then

N. NO

II.

<M>Ry = M Y f o r a l l y If y

Proof.

E

r

I ,

hence

'My. ~ ~

9.175


L e t N b e tame and M , ,

N o - s u b g r o u p s o f M w i t h MIM 2 . . . M k

<M1>R<M2>R.. -~ Proo

f.

.R

=

TO}

{Ol. Then

=

L e t k = 2 ( a n d t h e n p r o c e e d by i n d u c t i o n ) . Now b y 9 . 1 7 4 ,

to1 f o r a l l y i s f a i t h f u l , < M , i e , < M 2 > * , = {U}.

9.176

<M,>% M2y = M 1 M 2 y =

COROLLARY ( S c o t t ( 1 7 ) .

9.177

Proof.

n

E

<M2>Ry = I

1'.

Since

The sum L o f a l l n i l p o t e n t l e f t

N we g e t Ln

=

ELi".

i s a.1 i d e a l o f N .

0

E a c h Lin

i s an N - s u b -

g r o u p o f N and i s n i l p o t e n t , s i n c e N L c L . 1

R

i s a n i l p o t e n t l e f t i d e a l by 9.176,

< L 1. n j L C C L ~=

Nr

too.

o f a tame n e a r - r i n g N = N

For

Suppose

N i s n i l p o t e n t and i f N

5

i s nilpotent,

THEOREM ( S c o t t ( 1 7 ) .

i d e a l s Li

If M

N O

i s tame t h e n < M > R

be

a s we1 1 .

t h a t N i s t a m e on N T . =

. . . ,Mk

Hence whence

L. T h e r e f o r e L n S L .

These a r g u m e n t s a l s o show: 9.178

P R O P O S I T I O N (Lyons-Meldrum

N tame and B S A

(2)).

Nr b e u n i t a r y ,

Let

= N ~ .T h e n t h e i d e a l g e n e r a t e d b y B i n A

i s t h e same a s t h e o n e g e n e r a t e d i n

r

and

i s g i v e n by

CNB. BEB Another i n t e r e s t i n g f a c t i s the following. completely non-abelian o f an i d e a l o f over

A.

C a l l an N-group

r

i f each n o n - z e r o homomorphic image A

Nr i s e i t h e r n o n - a b e l i a n o r no n E N d i s t r i b u t e s

$9 MORE CLASSES OF NEAR-RINGS

356

I t f o l l o w s f r o m 2.23

distributive.

If

r

t h a t t h e i d e a l l a t t i c e o f NT i s t h e n

r

i s non-abelian and simple then

i s an

example o f a c o m p l e t e l y non-abel i a n I ( r ) - g r o u p . 9.179

I f ,,,T i s t a m e a n d c o m p l e t e l y

PROPOSITION ( K a a r l i ( 5 ) ) .

9QN

n o n - a b e l i a n t h e n L 1 y n L 2 y = ( L 1 L~2 ) y f o r a l l L , , L 2 and y

E

Proof.

I'. ( L 1 " L 2 ) y ~ L 1 y n L 2 yi s t r i v i a l .

Conversely,

if

a,B E L 1 y n L 2 y a n d n E N t h e n a + B - a - B a n d n ( a t g ) - n g - n a a r e i n ( L 1 ~ L 2 ) y ,s i n c e ( L 1 n L 2 ) y i s an No-subgroup,

consequently an i d e a l

o f N T , The l e f t i d e a l g e n e r a t e d b y a l l a+B-a-B a n d n ( a + B ) - nR - n n w i t h ~ , B E L , ~ A L a~n dY EN

Nr i s c o m p l e t e l y

coincides w i t h L 1 y n Lzy, since non-abelian. Now we

take

Hence L l y n L z y c ( L , o L z ) y .

a l o o k a t t h e s t r u c t u r e o f a tame n e a r - r i n g .

O b v i o u s l y we g e t 9.180

P R O P O S I T I O N I f NT i s tame t h e n NT i s 0 - p r i m i t i v e i f f i t

i s 2-primitive.

From t h i s i t l o o k s q u i t e p l a u s i b l e t h a t J o ( N ) f o r a tame n e a r - r i n g N. 9.181

22(N) holds

We w i l l r e t u r n t o t h i s q u e s t i o n l a t e r o n L e t N b e tame on N T , M


s NN a n d 0

r.

Then M+(O:y) a N. R P r o o f . By 2 . 1 5 , M + ( O : y ) 5 -Y E

=

n,n' E M ,

n(mY+n'y)

-

N. I f

m E M ,

a

E

(O:y),

NO

r,

n n ' y E M ~ , s i n c e My s 1

+

=

s2.

Obviously, if S is a group of automorphisms, this concept of fixed-point-freeness coincides with o u r well-known concept for automorphism groups (cf. e.g. 4.52). It can be shown that, if S is fixed-point-free a n d finite, S can be written as S = G , u... u G n u { G I , where G ,,... ,Gn are groups with identities e , ,.. . ,en a n d e . e . = 6 . (hence S i s a completely regular inverse 1

semigroup).

J

1 j

See also Maxson-Smith ( 1 1 ) .

§ 9 MORE CLASSES OF NEAR-RINGS

364 9.195

Let N E

THEOREM ( M a x s o n - S m i t h ( 8 ) ) .

n,

be a f i n i t e

n e a r - r i n g . T h e n N i s s e m i s i m p l e a l l o f i t s s i m p l e summands e i t h e r non-rings o r f i e l d s i f f

being

N i s isomorphic t o

some M S ( T ) , w h e r e S i s a f i x e d - p o i n t f r e e s e t o f e n d o morphisms o f

r

finite,

r.

More g e n e r a l l y ,

M S ( r ) , w i t h S S Endr,

i s semisimple i f f S i s a completely regular

i n v e r s e semigroup.

More i n f o r m a t i o n c a n be o b t a i n e d i f S i s s p e c i a l i z e d .

The f i r s t

c o l l e c t i o n o f r e s u l t s w h i c h we m e n t i o n c o n c e r n s a o n e - e l e m e n t set S = {s},

i n t h e second s e r i e s o f r e s u l t s , S i s a semigroup

o f " l i n e a r " maps. O f c o u r s e ,

MCsl(r)

=

M

Let

r

<S>

(r),

where i s > i s t h e

subsemigroup generated by s . 9.195

THEOREM ( M a x s o n - S m i t h ( 2 ) ) .

s =

E

End(r).

Then

be a f i n i t e g r o u p and

t h e f o l l o w i n g a s s e r t i o n s h o l d f o r N:

=

M{s,a3(T):

( a ) I f s i s n o n - n i l p o t e n t and n o t i n v e r t i b l e t h e n N i s n o t 2 - s e m i s imp1 e.

( b ) L e t s be n i l p o t e n t o f degree n > I , =

{f E NI m- 1

sn-'(y)

( y ) = 6 ) , b u t f o r n o yl

: s

w i t h maximal m. a n d N/2,(N) (c)

If

0 3 a n d A : = IS

=

r

T h e n a,(N)

r

E

E

L(y):=

ker slvy

E

r:

we h a v e s m ( y ' ) = E l

fl

L ( 6 ) = {f ~ N l f / *= U j . 6EA g M o ( A ) ; hence N/g2(N) i s s i m p l e . =

i s a v e c t o r space and s i s l i n e a r t h e n N i s

simple i f f N i s 2-semisimple.

9.197

THEOREM (Maxson-Smith ( 3 ) ) . identity, w i t h fr:

L e t R be a f i n i t e r i n g w i t h

RN a f i n i t e u n i t a r y R-module and S : = V + V,

v-frv.

( a ) MS(V) = I f E Mo(V)I ( b ) If R i s simple,

{ f r l r € R)

Then

V

r E R v m

E

M:

f(rm) = rf(m)l.

so i s M S ( V ) , and MS(V) i s a n e a r - r i n g

i f f R i s a f i e l d w i t h dimRV>l.

( c ) If R i s s e m i s i m p l e ,

w i t h n o n e o f i t s s i m p l e summands

b e i n g a f i e l d then MS(V) i s a r i n g . ( d ) If R i s n o t a f i e l d b u t i f M S ( V ) i s s t i l l s i m p l e t h e n

MS(V)

i s t h e r i n g EndR(V).

9h Bicentralizer near-rings

365

Now we t u r n o u r a t t e n t i o n t o t h e s t r u c t u r e o f M o ( r ) f o r some G G 2 A u t ( T ) . Even s t r o n g e r t h a n b e f o r e one c a n s a y t h a t MG(T) = =

M i n M(r). Up t o now, t h i s s i t u a t i o n was o n l y c o n s i d e r e d f o r l i n e a r a u t o m a t a / l i n e a r s y s t e m s , i n w h i c h c a s e N t u r n s o u t t o be a r i n g . The n o n - l i n e a r s i t u a t i o n a n d t h e u s e o f n e a r - r i n g s seems t o be m o s t promissing. T h e s e i d e a s w i l l be p u r s u e d i n f o r t h c o m i n g p a p e r s .

Our w o r l d i s b e c o m i n g i n c r e a s i n g l y c o m p l i c a t e d a n d t h e a u t o m a t a a n d s y s t e m s i n v o l v e d a n d a r i s i n g a r e i n many c a s e s f z r away f r o m b e i n g l i n e a r . B u t i n many c a s e s t h e s t a t e s e t s Q c a r r y a n a t u r a l g r o u p s t r u c t u r e ( e . g . Q = R n ) . Hence o n e m i g h t hope t h a t n e a r - r i n g s can be o f use i n t h e n o n - l i n e a r c a s e , t h u s becoming a n i m p o r t a n t t o o l i n t h e u n d e r s t a n d i n g of o u r world.

392

89


j ) MISCELLANEOUS TOPICS

I n t h i s f i n a l s e c t i o n we i n t e n d t o u i v e b r i e f d e s c r i p t i o n s o f t o p i c s we d i d n ' t d i s c u s s i n o u r j o u r n e y t h r o u q h t h e " n r . - u n i v e r s e " u n t i l now. A g a i n i t s h o u l d be n o t e d t h a t b e i n q i n t h i s s e c t i o n s h o u l d n o t imply any d i s c r i m i n a t i o n o f t h i s s u b j e c t ( a s b e i n o " l e s s i m p o r t a n t " ) . ble h a v e t o r e a c h a n e n d o f t h i s monooraph t h e r e a d e r rniflht b e t i r e d .

9 . 2 6 0 SEMIPRIMARY NEAR-RINGS w e r e i n t r o d u c e d _-

and studied by

K a a r l i in a s e r i e s of p a p e r s . N = No i s c a l l e d semiprimary i f N c o n t a i n s a f i n i t e c h a i n of i d e a l s such t h a t each f a c t o r i s e i t h e r n i l p o t e n t o r isomorphic t o a r i n g of l i n e a r t r a n s f o r m a t i o n s on a f i n i t e - d i m e n s i o n a l v e c t o r s p a c e o r i s o m o r p h i c t o a c e r t a i n r i n g o f homomorphisms. S e m i p r i m a r y

n r . ' s have t h e D C C N ; i f a semiprimary n r . N i s a r i n g then i t i s semiprimary i n t h e sense of ( J a c o b s o n ) ( i . e . N / Y ( N ) h a s t h e D C C L ) . In ( 7 ) , K a a r l i shows t h a t N i s s e m i p r i m a r y iff a,/2(N) i s n i l p o t e n t , the N-group N/$'1/2(N) has t h e D C C I a n d no N-group o f t y p e 2 i s N - i s o m o r p h i c t o o n e o f i t s p r o p e r f a c t o r N - g r o u p s . The s t r u c t u r e and t h e r a d i c a l t h e o r y o f semiprimary n r . ' s N and t h e i r N-groups was developed i n K a a r l i ( 2 ) , ( 4 ) , ( 6 ) and ( 7 ) ( a n d sometimes mentioned i n t h i s book). 9 . 2 6 1 T O P O L O G Y IN N E A R - R I N G S The s t a r t i n g p o i n t was B e i d l e m a n - C o x

( 1 ) which c o n t a i n s

d e f i n i t i o n s and s t r u c t u r a l p r o p e r t i e s o f t o p o l o o i c a l rings.

near-

T o p o l o s i c a ? n r . ' s on r e l a t i v e l y f r e e F r o u p s were c o n s i d e r e d by T h a r m a r a t n a n ( 3 ) ( s e e 6 . 3 5 ( f j ) . B e t s c h ( 3 ) c o n s i d e r s t o D n l o n i c 3 1 S D ? C C S i n d u c s d b y . ~ - n r-i mitive iaeais i v = i . 2 ) . Nr.'s o f c o n t i n u o u s m a p p i n q s on t o p o l o a i c a l q r o u p s ( t o t a l l y disconnected topoloqical aroups, Banach-spaces, r e a l n u m b e r s , . . . ) were c o n s i d e r e d by B e t s c h ( 3 ) ,

9j Miscellaneous topics

Magill ( 1 ) - ( 3 ) ,

Hofer ( 1 ) - ( 5 ) ,

Yamamuro ( 5 ) ,

393

Palrner-Yama-

muro ( I ) , B l a c k e t t ( 4 ) - ( 6 ) . Su ( 1 ) , ( 2 ) , Holcornbe ( 3 ) , ( 4 ) , H . D . Brown ( 2 1 , R . H o f e r ( 3 ) , ( 5 ) , S e p p a l a ( I ) , S u ( 2 ) a n d Adler ( 1 ) . Fbr i n s t a n c e , Yamamuro o b t a i n s t h e f o l l o w i n o r e s u l t i n ( 5 ) : Let B be a r e a l B a n a c h - s p a c e o f d i m e n s i o n 2 2 , a n d l e t

N b e a n r . o f c o n t i n u o u s m a p p i n n s B -+ B , containin4 Maff(B). Then e v e r y a u t o m o r p h i s m o f B i s i n n e r . T h i s implies t h a t i f Bl,?ll and B z , M z a r e two c o u p l e s a s % and N2 a r e a l s o t o p o a b o v e a n d Pi1 = PI2 t h e n P i 1 l o g i c a l l y isomorphic (homeonorphic). See Wefelscheid ( 1 ) , ( 2 ) and ( 7 ) f o r t o p o l o g i c a l n e a r - f i e l d s See Neuberger ( 1 ) , ( 2 ) f o r a p p l i c a t i o n s of n r . ’ s i n f u n c t i o n a l a n a l y s i s . S e e M a g i l l ( 9 ) f o r an e x c e l l e n t summary.

1.262 r l E A R - R I N G S

IN A L G E B P A ! C

TOPQLQGv

I~

I n decomposina p o l y h e d r a s one meets n e a r - r i n q s as s t r u c t u r e s w h i c h a n n i h i l a t e hornoloay a r o u p s ( s e e

(1)). C u r j e l ( 2 ) c o n t a i n s ( a m o n n o t h e r s ) t h e f o l l o w i nn r e s u l t s : Let A be a f i n i t e c o m p l e x , C A t h e r e d u c e d s y s p e n s i c n II t b e n e a r - r i n o (k!ith i d e n t i t y ) O F o f A and N ( 1 A ) Curjel

7 :

homotopy c l a s s e s o f b a s e - p o i n t p r e s e r v i n o s e l f m a p s o f T f i . U s i n g t h e induced endomorphisms o f H , ( Z A ) , t h e followincl a s s e r t i o n s c a n be shown t o b e e q u i v a l e n t : (a)

m,nEN:

nn-nm

i s o f f i n i t e additive order.

( b ) The g r o u p o f i n v e r t i b l e e l e m e n t s i n t h e n o n o i d

(N,*)

( = i t s qroup k e r n e l ) i s f i n i t e .

(c)

nEN:

n n i l p o t e n t =>

I f the Betti-numbers of

n i s o f f i n i t e additive order.

a r e knOVJn, one can d e c i d e w h e t h e r o r n o t ‘I b a s t h e s e o r o p e r t i e s b y a P e c h a n i c a l a p p l i c a t i o n o f t t i l t o n ’ s formclla f o r t h e c o n o t o p y a r o u p s o f d d n i o r . of S o h e r e s . A l s o , EA

89

394


9.263 V A L U A T I O I i TH_EORY O N FIEPiR-RI>JGS T h i s i s d e v e l o p e d i n Zenimer ( 3 ) , ( 4 ) and ( f o r n e a r - f i e l d s )

i n Wefelscheid ( 6 ) , ( 7 ) .

9.264 EXTENSIONS A N D H O M O L O G Y Maxson ( I ) , C h o u d h a r i ( I ) , ( ? ) , S e t h - T e w a r i (l), Mason ( 3 ) , ( 4 ) , B a n a s c h e w s k i - N e l s o n ( I ) , Oswald ( 7 ) , Maxson-flswald ( I ) , Meldrum ( 8 ) a n d P r e h n ( 1 ) - ( 3 ) c o n s i d e r e x a c t s e q u e n c e s of N-groups, i n j e c t i v i t y , p r o j e c t i v i t y a n d t h e c o n n e c t i o n s

t,o s e m i - s i m p l i c i t y ( s e e 5 . 4 9 , 5 . 5 0 a n d 9 . 1 5 5 ) . Steinegger ( 1 ) d e s c r i b e s extensions o f near-rings by s e t s of f u n c t i o n s ( s i m i l a r t o t h e r i n g c a s e ) . For d g n r . Is, h o m o l o g i c a l i n v e s t i g a t i o n s w e r e c a r r i e d o u t b y F r o h l i c h ( 5 ) - ( 8 ) ( " n o n - a b e l i a n homological a l g e b r a " ) ; c f . Lausch ( 1 ) , ( 3 ) a n d L o c k h a r t ( 4 ) . 9 . 2 6 5 NEAR-RINGS A N D CATEGORIES

L e t c be a c a t e q o r y w i t h f i n i t e p r o d u c t s and a f i n a l XEC' be a q r o u p o b j e c t . Then Mor(X,)o = ( c f . 1 . 4 ( a ) ) i s a nr. w i t h t h e obvious o p e r a t i o n s = M(X)

o b j e c t . Let

(Holcombe ( 3 ) , ( 7 ) , ( 8 ) ) . Holcomhe s t u d i e s t h e s e n e a r - r i n q s

i n v a r i o u s c a t e a o r i e s , Hom010nv a n d c o h o m o l o n y clroups c a n be v i e w e d a s c e r t a i n N - o r o u p s f o r some nr. N. S i m i l a r c o n s i d e r a t i o n s ( i n a d d i t i v e c a t e o o r i e s ) can be f o u n d i n H u q ( 1 ) a n d Ai j a z - H u q ( 1 ) . A categorical investigation t o radical theory i s i n Holcombe ( 7 ) a n d Holcombe-Walker ( 1 ) . In ( 1 5 ) , ( 1 6 ) , ( 1 7 ) , C l a y g i v e s a d e t a i l e d account on n r . ' s ( " f i b e r e d p r o d u c t n e a r - r i n g s " ) a r i s i n g i n t h e s t u d y of c a t e g o r i e s with g r o u p o r cogroup o b j e c t s . F r o h l i c h ( 4 ) - ( 8 ) s t u d i e d d g n r . ' ~b y means o f c a t e g o r i c a l c o n s i d e r a t i o n s . Mahmood ( 1 ) - ( 4 ) c o n t i n u e d t h e s e s t u d i e s and showed (among o t h e r r e s u l t s ) t h e s u r p r i s i n g f a c t s t h a t p r o d u c t s ( c f . 6 . 9 ( d ) and t h e f a c t t h a t t h e d i r e c t p r o d u c t of d g n r . ' ~i s not d . g . in g e n e r a l ! ) , coproducts, l i m i t s a n d c o l i m i t s e x i s t i n t h e c a t e g o r y of d g n r . ' s ( N , D ) ( w i t h


395

( N , D ) - h o m o m o r p h i s m s a s i n 6 . 1 7 a s m o r p h i s m s ) . MahmoodMeldrum ( 1 ) showed t h a t s e v e r a l c a t e g o r i e s a r e l i n k e d b y f u n c t o r s a r i s i n g f r o m d g n r . ' ~ . Mahmood-Meldrum ( 2 ) a p p l i e d s e v e r a l o f t h o s e i d e a s t o s t u d y s u b d i r e c t p r o d u c t s of dgnr. 's. 9 . 2 6 6 N E A R - R I N G S O N A G I V E W SEMIGROUP

I n t h i s s i t u a t i o n one s t u d i e s a problem " d u a l " t o t h e one s t u d i e d i n $9 d ) . Given a m u l t i p l i c a t i v e s e m i g r o u p ( N , . ) , w h i c h a d d i t i o n s + c a n be d e f i n e d o n N i n o r d e r t o t u r n (N,+,.) i n t o a n e a r - r i n g ( w i t h c e r t a i n p r o p e r t i e s ) . For i n s t a n c e , Ligh ( 2 0 ) c l a s s i f i e d a l l f i n i t e groups ( G , . ) such t h a t G , and a l l subgroups of i t , a r e m u l t i p l i c a t i v e groups of n e a r - f i e l d s . I t t u r n s o u t t h a t G i s e x a c t l y one o f t h e f o u r t y p e s : ( a ) Z n , such t h a t e v e r y d i v i s o r d of n i s o f t h e f o r m d = p"-I ( p a p r i m e ) , ( b ) t h e q u a t e r n i o n g r o u p o f o r d e r 8 , ( c ) a m e t a c y c l i c group of o r d e r 24, ( d ) a b i t e t r a h e d r a l g r o u p of o r d e r 2 4 . See a l l p a p e r s in t h e b i b l i o g r a p h y which a r e l a b e l l e d by M'. 9 . 2 6 7 CONDITIONS F O R N TO B E F I N I T E L i g h ( 1 ) has s h o w n t h a t i f N c o n t a i n s n r i g h t z e r o d i v i s o r s ( N I S n 2 , hence N i s ( a t l e a s t o n e o f them E N d ) t h e n

f i n i t e . See a l s o Linh-Malone ( 1 ) . F o r r i n g s , t h e D C C a n d A C C on s u b r i n o s f o r c e t h e r i n n t o be f i n i t e . B e l l - L i o h

(1) extended t h i s result t o

d g n r . I s and o b t a i n e d s i m i l a r o t h e r f i n i t e n e s s c o n d i t i o n s ( m a i n l y f o r d g n r . ' s ) . See a l s o Bell

( I ) , F e i g e l s t o c k ( 1 ) a n d John ( 1 )

( 3 ) , ( 1 1 ) , Bell-Liqh and c f . 9 . 2 6 5 .

9 . 2 6 8 RESIDUAL FINITENESS Call a n algebra A r e s i d u a l l y f i n i t e i f f o r a l l a , b E A , a+b, there i s a f i n i t e algebra A i n t h e v a r i e t y g e n e r a t e d by a ,b A and a homomorphism h : A + A with h ( a ) t h ( b ) . Free n e a r a ,b a r e r e s i d u a l l y f i n i t e ( a n d word p r o b l e m s i n rings in

no

396

§ 9 MORE CLASSES OF NEAR-RINGS

them a r e all free sidually of F D ,V

solvable). If i s a v a r i e t y o f g r o u p s i n which groups a r e r e s i d u a l l y f i n i t e a n d i f D i s a r e f i n i t e semigroup then t h e " f r e e d . g . n e a r - r i n g " 6 . 2 1 i s r e s i d u a l l y f i n i t e , t o o . See John ( 1 ) .

9 . 2 6 9 NON-ASSOCIATIVE NEAR-RINGS

In R a m a k o t a i a h - S a n t h a k u m a r i ( 2 ) , ( 3 ) a n d S a n t h a k u m a r i ( I ) , zero-symmetric l o o p n e a r - r i n g s N a r e s t u d i e d ( w h i c h means

t h a t ( N , + ) i s a l o o p ) . Loop n r . ' s a r i s e f r o m t h e s t u d y o f m a p p i n g s o f a l o o p i n t o i t s e l f ( c f . 1 . 1 1 8 ) . Among o t h e r r e s u l t s , the a u t h o r s o b t a i n e d a d e n s i t y theorem f o r v-primit i v e l o o p n r . ' s s i m i l a r t o 4 . 3 0 . C f . a l s o 8 . 4 1 and 8 . 4 2 . Timm ( 5 ) - ( 7 ) s t u d i e d m u l t i p l i c a t i v e l y n o n - a s s o c i a t i v e n e a r r i n g s . C f . 8.48. See a l s o S t e f a n e s c u ( l ) - ( l O ) . 9 . 2 7 0 COMMUTATORS. DISTRIBUTORS A N D SOLVABILITY D i s t r i b u t o r s a r e d e f i n e d i n 9 . 7 9 . For a d e t a i l e d s t u d y o f t h e s e c o n c e p t s s e e Esch ( 1 ) a n d c o n f e r H . D . Brown Esch ( 1 ) a l s o c o n t a i n s r e s u l t s due t o F r o h l i c h ( 1 ) , ( 2 ) on d i s t r i b u t o r s a n d "weak d i s t r i b u t i v i t y " i n d p n r . ' s (cf. 6.16).

S e e a l s o Mason ( 1 ) , ( 2 )

a n d Maxson ( 1 ) .

N r . ' s g e n e r a t e d by t h e c o m m u t a t o r s o f a ( n o n - a b e l i a n ) g r o u p a r e s t u d i e d i n Gupta (1). S e e a l s o Curjel ( 1 ) . D a s i c ( 1 ) - ( 9 ) , D a s i c - P e r i c ( I ) , K u z ' m i n ( 1 ) . Meldrum ( 1 3 ) , Oswald ( 1 ) , ( 5 ) , R o b e r t s ( 1 ) a n d S c o t t ( 7 ) . 9 . 2 7 1 DISTRIBUTIVE NEAR-RINGS T h i s i s t h e p l a c e where t h e t h e o r i e s o f n e a r - r i n o s a n d s e m i r i n o s m e e t . !,!e m e n t i o n e d t h e s e n r . ' s a l r e a d y i n 1 . 1 5 , 1 . 1 0 7 and 1 . 1 0 8 . A l l o f 96 i s a p p l i c a b l e . T a u s s k y ( 1 ) a l s o showed t h a t i n a d i s t r i b u t i v e n r . rl e i t h e r e a c h e l e m e n t

i s a zero divisor o r N i s a rino. A simple d i s t r i b u t i v e

n r . i s a l s o a r i n q ( F e r r e r o ( l ) , Lirlh ( 1 3 ) ) . F o r more d e t a i l s s e e H e a t h e r l y ( 4 ) , ( 6 ) , H e a t h e r l y - L i g h ( I ) , H e a t h e r l y - O l i v i e r ( 3 1 , L i g h ( 8 ) , ( 1 5 ) , Malone ( 7 ) a n d ( a u n i f y i n g p r e s e n t a t i o n ) Weinert ( 7 ) - ( 1 0 ) .


N

i s s a i d t o be n - d i s t r i b u t i v e

\I

a b e l i a n and

=

(ncIN)

.

- 1 xyizi

1=1

n-distributive for a l l

neIN

i s a nr.

N is

if

Un(N)

with entries

N t o g e t h e r w i t h t h e u s u a l a d d i t i o n and

iff N i s n-distributive.

f o r m a l power s e r i e s ,

"Gaussian n e a r - r i n o s

N(i)".

Ligh (17))

Also,

Hn(P!)

one can s t u d y

o r o u p n e a r - r i n n s and

These s e t s a r e ( u n d e r t h e

usual operations) always near-rinos distributive

i s

.

m u l t i p l i c a t i o n then (Heatherly ( 4 ) , polynomials,

(N2,+)

N i s p s e u d o - d is t ri butive

I f one c o n s i d e r s t h e n x n - m a t r i c e s f r o m some n r .

if

, . . . ,yn,zl ,... ,zncf4:

x,yl

n

n x ( . c yizi) 1=1

397

i f fil i s p s e b d o -

( s e e H e a t h e r l y - L i a h ( 1 ) f o r t h i s a n d many

other results concerninn pseudo-distributive

near-rinns).

C o n f e r a l s o B e i d l e m a n (1) a n d S u p t a ( l ) , as w e l l a s 9.16'3. Sieno-Stefan0

( 1 ) showed t h a t a l l

2v c o i n c i d e

in a distri-

butive nr. F o r t h i s and 9.271, D'

9.272 -

see a l s o a l l o t h e r papers marked by

b

and

i n the bibliography.

CHARACTERIZING SERIES

L e t NT be a u n i t a r y N-group.

An N - s e r i e s

o f NT i s a s e r i e s

o f l e n g t h n: r = r o z r , z . . . 2 r n = Co) w i t h 4 -N r i for each i < n. I f I 9 N t h e n t h i s s e r i e s i s s a i d t o be a n-I charactsrizing series f o r I i f I= (Titl:Ti) and i=l Iris for 0S i n-2. I has a c h a r a c t e r i z i n g s e r i e s only i f

I k s i ( o : r ) f o r some k

E

IN.

A l l characterizing

s e r i e s f o r I 2 N h a v e t h e same l e n g t h n , a n d n i s j u s t t h e nilpotency class of I/(o:r).

F o r t h i s a n d many o t h e r r e -

s u l t s see Lyons ( 7 ) , Lyons-Meldrum ( 1 ) , ( 2 ) 9.273

CENTRAL N - S U B G R O U P S

and Meldrum ( 7 ) .

are studied i n Scott (22).

A sN r i s c e n t a l i f n i s c o n t a i n e d i n t h e c e n t e r o f ( r , + ) and n E N v I15 c A : n ( y t 6 ) = n y + n S . I f A i s c e n t r a l t h e n A iN r . I f r = T I b r 2 and r sN r has i n t e r s e c t i o n { o l w i t h r l a n d r 2 t h e n A i s c e n t a l . I f a r a~n d N r 2h a v e c e n t a l N-subgroups A1 ,A2

with Al

I(

gN

A2

by h then

rl

@

T2/A

with

89

398


3 : = { ( 6 1 , h ( ~ , ) ) ( 6 1 c ~ 1 )i s c a l l e d a c e n t r a l p r o d u c t o f

and

r2.

r

If

r,

zN

TI

t h e n any N-homomorphic image o f

%I

Nr i s a c e n t r a l p r o d u c t . 9.274

C-Z-TRANSITIVE

A N D C-Z-DECOMPOSABLE N E A R - R I N G S

N i s "C-Z-transitive"

\r

nccFIP

i f

nZEI.I C

3

noEr10 :

n O n c = n;.

ri0 N c

I n t h i s case,

"C-Z-decomposable"

i f

related t o a.a.n.r.'s Heatherly (2) rinqs.

9.275

N

4

(these nr.'s

are closely

!).

developes an i d e a l t h e o r y f o r t h e s e near-

N

N E A R - R I N G S w e r e a l r e a d y t o u c h e d i n 9.122. N i s H-monogenic

0'

hlh2 = 0

(i.e. H =

=

PIc

also P i l z (1),(6).

Cf.

H-MONOGENIC

If HsN

i s s t r o n o l y monoclenic. N i s

*

2

if N g

H and H i s " i n t e g r a l "

hl=O v h 2 = 0 ) . I f N i s H - m o n o g e n i c w i t h

{ O l t h e n N h a s z e r o m u l t i p l i c a t i o n . On t h e o t h e r h a n d ,

i f N i s N-monogenic t h e n N i s i n t e g r a l .

H-monogenic n e a r -

r i n g s can be c o n s t r u c t e d b y a g e n e r a l i z a t i o n o f F e r r e r o ' s method p r e s e n t e d i n 1 . 4 ( b ) . Heatherly-Olivier

F o r t h i s and o t h e r t o p i c s see

( 3 ) and O l i v i e r ( 1 ) , ( 2 ) .

9 . 2 7 6 N-SYSTEMS ___

A nr.

w i t h r i o h t c a n c e l l a t i o n law and a " h a l v a b l e

t.lEr)O

idempotent

e

+ 0"

(i.e.

3

hEN:

h+h = e )

i s called

N - s y s t em. Every ii-system i s a b e l i 8 n (see t h e proof o f 9.13(b)) inteqral

(so 3b)Z)

near-field,

and

i s a t hand), A f i n i t e ri-system i s a

b u t t h e r e do e x i s t i n f i n i t e H-systems w h i c h

are neither rinqs nor near-fields

(see Linh-f?alone ( l ) ,

L i g h - M c Q u a r r i e - S l o t t e r b f c k ( 1 ) a n d U t C u a r r i e (1),(3)).

If

N IMo(r)

a n d ti i s a n : I - s y s t e m c o n t a i n i n o

e v e r y f u n c t i o n o f N i s odd ( c f .

9.152jb)).

id,,

then


9.277

399

AUTOMORPHISM G R O U P S OF N E A R - R I N G S Scott (18) studied the group Aut(N) o f a l l automorphisms o f a n e a r - r i n g N t h e n an : x A

+

nxn-'

n invertible)

i s the

m o r p h i s m s o f N. where D(N)

=

~ n , I.f

i s i n Aut(N).

(near-ring-)

n t Nd i s i n v e r t i b l e

I n n ( N ) : = { a n I n e Nd

A

(normal) subgroup of a l l i n n e r auto-

As f o r g r o u p s we g e t D ( N ) / Z ( N ) - I n n ( N ) ,

Cn € N d l n i n v e r t i b l e ] a n d Z ( N ) = i n

E

D(N)lnn=id].

N i s c a l l e d complete

I n another analogy t o groups, a n r .

( c f . 9 . 1 0 0 ( b ) ) i f Z ( N ) = { I } , N &?Io and i f a l l autornorphisms of N a r e i n n e r . I f r i s a complete group and N = A ( r ) such t h a t ,,,T i s m o n o g e n i c t h e n N i s a c o m p l e t e n e a r - r i n g w i t h Aut(N)-Aut(r). By 7 . 1 6 ,

7.59

For instance,

I(Sn],

nf6,

i s o f t h i s type.

r.

M o ( r ) i s c o m p l e t e and A u t ( M o ( r ) )

See a l s o

and 9.226.

Magill

(7) studies a nr.

c a l l e s N,

:=

(N,+,.,)

(N,+,.),

with

n

m

c h o o s e s some a E N a n d rn : = nam

a

the near-ring

l a m i n a t e d by a. For N = M (IR) , Aut(Na) i s determined. cont I n a more g e n e r a l frame, automorphisms a r e s t u d i e d by Ncbauer ( 1 2 ) and P l o t k i n ( 3 ) . 9.278

DICKSON-NEAR- RINGS

The d e f i n i t i o n s o f c o u p l i n r l maps, Dickson n r . ' s

derived nr.'s

c a n be f o u n d i n 9.90.

and

For a detailed study

o f t h e s e c o n c e p t s s e e F l a x s o n ( 8 ) a n d Timm (6),(7).

O f course,

a Dickson near-rina

One may w r i t e a DFIR a s a r i n g and

(D,+,o)

Maxson shows e.q.

iff

(D,+,-)

(=: DNR) i s abelian.

the derived nr. i n (8) t h a t

has one and

(0

(O,+,o)

dcD":

(D,+,-)

where

(D,+,.,o),

bd

=

G

i s

6).

has an i d e n t i t y

$

6.

A finite

DNR w i t h i d e n t i t y i s a n f . . The i d e a l s t r u c t u r e o f a DiIR i s a l s o c o n s i d e r e d by P i e p e r (1) i n c o m p a r i n q t h e l e f t i d e a l s o f

(D,+,o). and

(D,+,-)

The c o n n e c t i o n b e t w e e n homomorphisms o f

(D,+,o)

and (O,+,*)

a r e s t u d i e d i n Maxson (13).

Kerby ( 5 ) s e t t l e s t h e q u e s t i o n i n which cases t h e nr. quotients o f

(D,+,o)

o f quotients o f

i s a !lickson

(D,+,*).

Of

o n e w.r.t. t h e r i n n


400

Aside from these considerations, "changed m u l t i p l i c a t i o n s " .

9.279

M a g i l l (2!,(7)

also studies

See a l s o 9 . 2 7 7 .

N E A R - R I N G S A N D N U M B E R THEORY

C o n n e c t i o n s between n e a r - r i n g s o f f o r m a l power s e r i e s and number t h e o r y were p o i n t e d o u t b y F r o h l i c h ( 9 ) .

Other

c o n n e c t i o n s a r e e s t a b l i s h e d i n Mazzola ( 1 ) and L i g h ( 2 0 ) . 9.280

NEAR-VECTOR S P A C E S I t seems n o t t o b e q u i t e c l e a r how t o d e f i n e a n e a r - v e c t o r

space.

Beidleman ( 1 ) d e f i n e d i t as a 2 - s e m i s i m p l e N-group

(N a n f . ) ,

and developed a k i n d i f " n e a r l y - l i n e a r "

algebra.

O t h e r a p p r o a c h e s t o t h i s c o n c e p t a r e made b y A n d r @ ( 3 ) ¶ ( 5 ) ¶

( 6 ) , Bachmann ( 2 ) .

Hule-Muller

(1) study a l q e b r a i c equations

over n r . ' s . 9.281

p-SINGULAR N E A R - R I N G S

SYLOW-TYPE T H E O R E M S ; f e r r e r o (1),(2)

N

and

\ : i \ = rn,

shows t h a t

k p /m

but

p

k+l

%in

Nd

implies the existence o f a two-sided i n v a r i a n t k subqroup o f N o f o r d e r p If I N / = p - q (p,qc_IP, p < q )

.

and

N

i s

n o t a b e l i a n t h e n ii h a s n o s u b n e a r - r i n o o f o r d e r p .

I f ti i s f i n i t e a n d

pclP,

/NI,

properly divides

:I

i s called p-singular

b u t II h a s n o s u b n e a r - r i n o w h o s e

o r d e r i s d i v i s i b l e b y p . So p - s i n a u l a r n r . ' s f o r n o t f u l f i l l i n o the Sylow-theorems".

N

i s

~ ? 7 a ~n d

p,N

if p

a r e "minimal

A p-sinnular nr.

i s s t r o n o l y monooenic.

See F e r r e r o (4)¶(5),(7)¶(18),(19) a n d S c o t t ( 8 ) . 9.282

LOCAL N E A R - R I N G S i s c a l l e d l o c a l i f L : = L(t!): = I X E N J X has no Nc7lofi??, l e f t i n v e r s e } srJ H . ( t h i s h a p p e n s i f f L i s a s u b q r o u p ) . Maxson ( 1 ) , ( 3 )

A local nr.

shows:

i s indecomposable.

w i t h DCC i s s i m p l e . A n r .

N i s l o c a l i f f N has a unique L i s q r . a n d i f PI i s n o t So L 222 - ixr 2-_I 226 1, T 227 1

rxl

0.

231

221

466

APPENDIX

*t 276 C,(N) 277

e Y LA

367 369

BIBD

T*

375

279 282

IFP (Po) nid.

y

. . . ,( P 4 ) 310

Ma(r) a.a.n.r.

313

%

316

cb C(i,k)

325

Dm 3 29 323

Am,

Dzn

n'

334

f.0.

334

7

5

309

k0

Near-rings : the theory and its applications, Volume 23 (North-Holland Mathematics Studies)

Design Theory: Volume 2 (Encyclopedia of Mathematics and its Applications)

Theory of Plates, Volume Volume II (Studies in Mathematics and its Applications)

Subdifferentials: Theory and Applications (Mathematics and Its Applications)

Discrete Mathematics and Its Applications

Superanalysis (Mathematics and Its Applications)

Discrete Mathematics and Its Applications

Discrete Mathematics and its Applications

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Advanced Number Theory with Applications (Discrete Mathematics and Its Applications)

Advanced Number Theory with Applications (Discrete Mathematics and Its Applications)

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Mathematical Methods of Game and Economic Theory (Studies in Mathematics and Its Applications Volume 7)

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Supermanifolds and Supergroups: Basic Theory (Mathematics and Its Applications 570)

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