Lectures in Rings and Semigroups

Lecture Notes in Mathematics Edited by A. Dold, Heidelberg and B. Eckmann, Z~Jrich

380 Mario Petrich Pennsylvania State University, University Park, PA/USA

With an Appendix by Richard Wiegandt

Rings and Semigroups

Springer-Verlag Berlin • Heidelberg • New York 19 74

AMS Subject Classifications (1970): 20-02, 20-M-20, 20-M-25, 20-M-30, 22-A-30, 16-02, 16A12, 16A20, 16A42, 16A56, 16A64, 16A80

ISBN 3-540-06730-2 Springer-Verlag Berlin • Heidelberg • New York ISBN 0-387-06730-2 Springer-Verlag New Y o r k . Heidelberg • Berlin This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher. © by Springer-Verlag Berlin • Heidelberg 1974. Library of Congress Catalog Card Number 74-2861. Printed in Germany. Offsetdruck: Julius Beltz, Hemsbach/Bergstr.

INTRODUCTION

Semigroup theory can be considered as one of the more successful offsprings of ring theory.

The relationship of these two theories has been a subject of particu-

lar attention only w i t h i n the last two decades and has generally taken the form of an investigation of the m u l t i p l i c a t i v e semigroups of rings.

The first and still

the most fundamental w o r k in this direction is due to L.M. G l u s k i n who certain dense rings of linear transformations These Lectures

represent an attempt

from the m u l t i p l i c a t i v e

studied point of view.

to put selected topics concerning both rings of

linear transformations and abstract rings, as well as their m u l t i p l i c a t i v e semigroups,

into a form suitable

for presentation

to students interested in algebra.

The Lectures are divided into three parts according to the clusters of covered topics.

part I consists of a study of certain semigroups and rings of linear transformations on an arbitrary vector space over a division ring. linear transformations two phenomena,

For dense rings of

containing a nonzero linear t r a n s f o r m a t i o n of finite rank,

from the present point of view,

are of decisive importance:

(a) its

m u l t i p l i c a t i v e semigroup is a dense extension of a completely 0-simple semigroup, and

(b) it has unique addition.

can be obtained by considering naturally

Because of (b), most

their m u l t i p l i c a t i v e

information about these rings

semigroups alone.

This leads

to a study of semigroups of linear transformations and in particular of

those satisfying

(a) above.

Hence in many instances we first estab]ish the desired

result for semigroups and then specialize

it to rings of linear transformations.

Even though the guiding idea adopted here is that first expounded by Gluskin the principal references are the books of Baer

[i] and J a c o b s o n

part II contains an investigation of various abstract rings, izations and representations.

[4],

[7]. their character-

The classes of rings under study here are semiprime

rings with minimal o n e - s i d e d ideals subject to various other restrictions. each of these classes of rings a m u l t i p l i c a t i v e b e i n g possible in view of their unique addition.

characterization

For

is provided,

this

Their m u l t i p l i c a t i v e semigroups

are either dense extensions of completely 0-simple semigroups or of their orthogonal sums.

Again some of the basic ideas here stem from G l u s k i n

reference is Jacobson

[4], but the m a i n

[7].

part III represents

a topological treatment of a left vector space, and its

ring of linear transformations, topology of modules and rings,

in duality with a right vector space.

Linear

linear compactness of vector spaces, various

IV

topologies on certsin rings of linear transformations, s topological vector space are covered here. appear here as topological rings. are due to Dieudonn~ and K~the

as well as s completion of

C e r t a i n of the rings studied earlier

Many of the basic

[2], but the chief references

ideas discussed in this part

sre the books of J a c o b s o n

[7]

[i].

The appendix contains s concise exposition of some principal achievements in the theory of linesrly compact modules and semisimple rings. consist of several characterizstions rings.

The chief original

The two main results

of linesrly compact primitive and semisimple

reference here is Leptin

12}.

Jacobson's Density Theorem

is included with a short proof.

The m e t h o d of m a x i m u m exploitstion of the m u l t i p l i c a t i v e often mskes

the use of various hypotheses more

that, st least for the rings under study here,

structure of a ring

transparent and demonstrates

the fact

the sddition is essentially extrane-

ous. The sources of these Lectures are numerous: references,

in addition to the above m e n t i o n e d

they include s vsriety of papers which sre generslly referred to in the

text.

There is slso a generous

time.

Among these are the results concerning the structure of simple rings with

minimal one-sided

sprinkling of results published here for the first

ideals in terms of Rees matrix rings,

and some related results due to Dr. E. Hotzel; author.

isomorphisms of the latter,

the remsining ones are due to the

For the sake of clarity and u n i f o r m i t y of presentstion,

have been rephrased and several new proofs hsve been provided. exercises at the end of most sections;

many known results There are several

they are designed to test the u n d e r s t a n d i n g

of the m a t e r i a l and sometimes extend the subject covered in the text. msterial

However,

the

in the msin body of the text is independent of exercises.

Part I was the subject of a one semester course in linear algebra in the Summer of 1969;

the entire Lectures

formed the content of a two semester course in topics

in ring theory in the school year 1971/72, both at the Pennsylvania State University. I sm indebted to Dr. D.E. Zitsrelli

for taking the notes

for Part I, to

Mr. JoJ. S t r e i l e i n for taking the notes for parts II and III, supplying the appendix,

and to Professor B.M. Schein

suggesting several improvements.

I am grateful

to include his u n p u b l i s h e d results, many slips,

to Dr. R. W i e g a n d t

for

for reading the m s n u s c r i p t and

to Dr. E. Hotzel

for the p e r m i s s i o n

to students in the two classes for correcting

as well as to all other persons who c o n t r i b u t e d

to the existence of these

Lectures. Statements in the text are referred to only by number: the same part,

the Arabic numerals are used,

if the statement is in

say 5.6 which is statement

6 in

V

Section 5; if the statement is in a different Part, numeral in affixed,

the number of the Part in Roman

say 1.5.6.

Since the w o r k of G l u s k i n on the subject at hand has provoked considerable interest,

it is hoped that a systematic and s e l f - c o n t a i n e d e x p o s i t i o n will propa-

gate the existing knowledge and stimulate new research in this highly promissing area of the r i n g - s e m i g r o u p cooperation.

TABLE OF CONTENTS

PART I SEMIGROUPS AND RINGS OF LINEAR TRANSFORMATIONS I.i

Definitions and notation

l

1.2

Dense rings of linear transformations

1.3

One-sided ideals of

1,4

Ideals of

1.5

Semilinear isomorphisms

34

1.6

Groups of semilinear automorphisms

47

1.7

Extensions of semigroups and rings

54

g(V)

Su(V)

7 16

and principal factors of ~ u ( V )

27

PART II SEMIPRIME RINGS WITH MINIMAL ONE-SIDED IDEALS II.l

prime rings

11,2

Simple rings

74

11.3

Maximal prime rings

85

11.4

Semiprime rings

91

65

95

ii .5

Semiprime rings essential extensions of their socles

11.6

Semiprime atomic rings

102

11.7

Isomorphisms

105 PART III

LINEARLY TOPOLOGIZED VECTOR SPACES AND RINGS III.I

A topology for a vector space

112

III.2

Topological properties of subspaces

120

III.3

Topological properties of semilinear transformations

125

111,4

Completion of a vector space

128

111.5

Linearly compact vector spaces

131

111.6

A topology for

£u(V)

136

III.7

A topology for

Su(V)

139

111.8

Another topology for

III.9

Complete primitive rings

£u(V)

143 146

VIII

APPENDIX ON LINEARLY COMPACT PRIMITIVE AND SEMISIMPLE RINGS O.

Introduction

152

i.

More about primitive rings

153

2.

Inverse limits and linearly compact modules

156

3.

Linearly compsct primitive rings

159

4.

Linearly compact semisimple rings

162

CURRENT ACTIVITY

167

BIBLIOGRAPHY

168

LIST OF SYMBOLS

175

INDEX

177

PART I

SEMIGROUPS The subject transformations dense

rings

finite mations one,

of

this part

rank h a v i n g

all o n e - s i d e d

vector

transformations

are c h a r a c t e r i z e d

of finite

ideals

are

in a vector

found,

the semigroup

taining

all

linear

duality w i t h

transformations

the given one

the u n d e r l y i n g vector

space

vector

of rank

is e x p r e s s e d

spaces.

are discussed.

in semigroups

representation

two semigroups

and rings with

part

space

of

In particular,

i with

linear

transforthe given

set of idempotents factors

an adjoint

is con-

each

are

con-

space

in

transformations

of

automorphisms

on a

of semilinear

of ideal

semigroups

is

ideals

in a vector

a discussion

to certain

all

transformations

of semilinear

of groups

of

in duality with

transformations

linear

of linear

transformation

for its principal

ends with

application

ring.

linear

ordered

linear

by means

A number This

and rings

For the ring of all

its partially

and ring of all

between

a division a nonzero

ways.

and a Rees m a t r i x

Isomorphisms

space over containing

semigroups

in several

characterized,

found.

TRANSFORMATIONS

an adjoint

structed.

For

OF L I N E A R

is a study of certain

on an a r b i t r a r y

of linear

rank

AND RINGS

extensions

and rings

of linear

transformations. The emphasis Thus many here

here

statements

for semigroups

is one

which

the m u l t i p l i c a t i v e

are usually

and are

then s p e c i a l i z e d

I.i A division all

the axioms

a commutative by

4.

The

ring

division

symbols

Throughout Denote

+,

this

the elements

case Roman

(or a skew

for a field

letters

in such a way

pair

that

is a field have

let

L

by lower

v,x,y, ....

is a function m a p p i n g The ordered

~

possibly

~X V (&,V)

under first

study. proved

AND N O T A T I O N is an algebraic

commutativi~y

system

satisfying

of m u l t i p l i c a t i o n .

and conversely.)

be a d i v i s i o n case G r e e k that

It will

V, say

postulates

ring and

letters, ~

be usually

(Thus denoted

V

be an a b e l i a n

and those of

acts on

V

V

by

group.

lower

on the left if there

(o,x) ~ ox.

is a left vector

the following

are

their usual meaning.

We say

into

of the rings

for these rings

to rings.

field or a sfield)

-, i, 0

section, of

DEFINITIONS

except

ring

structure

established

space

if

&

are fulfilled:

acts on

V

on the left

O ( x + y) = o x + o y

(oE 4, × , y E

v)

(O+~)x

(o,~ E a, x E

V)

~(~x)

= Ox+~x

= (o~)x

(x ~ V).

iX = x

The action and

those of If

~

is called

V

acts on

V

on the right,

postulates,

then

case,

using

the n o t a t i o n

Hence

the name v e c t o r

We will w r i t e

"V

Let called

the ordered

V

(left or right) the phrase

scalar m u l t i p l i c a t i o n ,

(V,4)

space

instead vector

pair

(&,V)

and

makes

of

(~,V)

(4,U)

linear

(~,V)

&

called

form

The set of all to be an abelian

linear

x(f+g)

group

V The

set

as follows. V*

V*

on the right

function

addition

V*

group

U

(&,V).

(V*,&)

forms on

for

is a right vector

Elements

of

V*

of confusion.

is to be regarded

group).

In such

A function

ss a

a case

a:V ~ U

is

V). spaces

apace

as operators

(&,A+) ~x of

of

(~,V)

where

on the right

on

into

4+

signifies

is the m u l t i p l i c a t i o n (4,V)

or functional)

into

(&,~+)

(&,U)

is easily

seen

by

(&,V)

f E V*

(i)

the usual

group

addition

can be made under

and

into

addition

o E &,

fo

of

the a b e l i a n

of homomorphisms. a right v e c t o r

(i).

We make

be defined

g

space act on

by

( x E v). space,

called

will be denoted

by

is

(~,V).

group of all h o m o m o r p h i s m s

under

×(fo) = (xf)o Then

act on the right.

danger

(x ~ V).

is an abelian

by letting,

In this

a,b,c, ....

defined

of the abelian

of linear

sp~ce.

if

transformation

= xf+xg

the abelian

First

A linear

transformations

under

it is a subgroup into

of left vector letters

as a left vector

(or a linear

group

(~,U)

and the scalar m u l t i p l i c a t i o n

o E 4, x E g+).

a linear

In fact,

itself &

V

spaces.

into

of the above

(x,y E V)

case Roman

g

group of

(here

scalars

4" is customary.

(o E 4, x E

transformations

lower

space over

be two left vector of

that

than just an abelisn vector

= xa+ys

We can consider

in

are called

a right vector

cases without

if it is clear

(rather

transformation

them by

the additive

A

that the scalars

in most

(ox)a = o(xa) We write

of

the right v e r s i o n

is called

it clear

can be used

space

(x+y)a

satisfying

(V,&)

is a (left or right)

a linear

and denote

elements

are called vectors.

the dual f,g,h,....

(or conjugate)

space of

3 A semigroup (usually

is a n o n e m p t y

mations

of

sition

(ab)x = a(bx)

written

on the right,

resulting

X

into itself,

semigroup

A linear (&,V).

written

(x C X),

on the left,

to be denoted

the c o m p o s i t i o n

will be denoted

transformation

of

g(V).

The same

the rin$ o f e n d o m o r p h i s m s

(&,V)

into

of

(~,V)

X,

binary

is a s e m i g r o u p

by

g(X).

operation

the set of all transforunder

the compo-

If the t r a n s f o r m a t i o n s

x(ab)

= (xa)b

(x E V)

and

are the

~(X). itself

(g,V) of

is called

of

~I(V);

and will be denoted

the addition

and will

an e n d o m o r p h i s m

is s s u b s e m i g r o u p

(g,V),

set endowed with of

set

is given by

by

the > e m i g r o u p o f e n d o m o r p h i s m s

or simply

an associative

For any nonempty

The set of all e n d o m o r p h i s m s

is called

£(V)

set together w i t h

called m u l t i p l i c a t i o n ) .

(i) forms

be denoted

by

by

a ring,

£(&,V)

of it

g(&,V) called

or simply

.

We have

seen that

can be given can be made made

If

U

transformations

group.

space over

4,

of

In addition, and for

(~,V)

into (~,U) + U = ~ , this set

for

U = V

the same

set can be

or a ring.

is a subgroup

is called

linear

of an abelian

into a right vector

into a semigroup

(&,U)

the set of all

the structure

of

a subspace

V

closed

of

(g,V).

under m u l t i p l i c a t i o n It is clear

that

by scalars

(~,U)

in

4,

then

is itself a vector

space. As in the case of a finite a linearly

independent

a generating and a basis

set

(every v e c t o r

of a vector

as a m i n i m a l

Any

have

two bases

denoted

to a basis

of

dim V.

every basis

To every

every vector

in

a complement

properties require proofs If mation

V

A

of a v e c t o r

further

V of

ranse of

and V

of

of

into

are U,

methods

which

then

in the form

V, and we write express

reference.

over

Va = {xa Ix E VJ

V

can be extracted.

to a basis of with

Most Zorn's

a

of the whole V

such

that

x E A, y E B;

We will

use

these

of these statements lemma).

For

is a linear

is a subspace

a, d e n o t e d

of

by rank a.

the null

V,

can be completed

the

at the end of this section. 4, and

called

of

V = A~B.

(axiom of choice,

spaces

the d i m e n s i o n

B

,

it is set

a basis

x+y

..

independent

set of vectors

a subspace

independent),

combination

has a basis;

linearly

can be completed

written

of

linear

set of vectors

one defines

is linearly

is by d e f i n i t i o n

there exists

is the rank of

is a subspace

always

independent

see the references

left vector

a; its d i m e n s i o n

N a = ix E V I xa = 0}

in

space w i t h o u t

discussion

U

A

subset

as a finite

space

generating

V,

space over a field,

finite

set or a maximal

linearly

can be uniquely

proof by transfinite and

A vector

of a subspace

subspace

B

is called

Any

V, and from every

In particular, space.

V.

generating

the same c a r d i n a l i t y

by

vector

(every

can be w r i t t e n

space

characterized

usually

dimensional

set of vectors

transfor-

U, called The set

space of

a.

the

Linear written space;

the r a n g e

space

are

a given

a function

left;

the null

vector

the dual

space

of a l i n e a r

pair

~:VX

(U,g)

U ~ ~

and

(&,V)

is c a l l e d

are d e f i n e d vector

=

(Vl,U)

+

(v2,u),

(v,ul+u2)

=

(V,Ul)

+

(v,u2) ~

~(v, u) =

of a r i g h t

and a left v e c t o r

form

u , u l , u 2 t U, o ~ &,

form

is n o n d e g e n e r a t e

and

(v,u)

stands

= 0

for all

u E

U

implies

that

v = 0,

vi)

(v,u)

= 0

for all

v 6 V

implies

that

u = 0.

In such

a case we

say

the d i v i s i o n

variously sizing

use

ring

Property (v,u)

is m o r e Let

over

(U,&,V)

LEMMA°

or

we

where

Oil

vector

should

~(v,u).

spaces

write

according

assumed

(U,&,V;~);

to the n e e d

as given).

is a left

(or a dual

Note

and

~l

that

pair)

we w i l l for emphain J a c o b s o n

is a r i g h t

vector

to: v E V

implies

for a p p l i c a t i o n ;

a vector

of dual

(U,V)

tacitly

for all

space.

a t-subspace)

If

by

t-subspace PROOF. is left

that

u = u ~,

analogously

A subspace

U

if for e v e r y

for v). of

V*

is c a l l e d

0 ~ x E V,

there

a total

exists

f ~ U

i__s_s!

pair,

vf u = (v,u) (V*,g).

t->ubspace

form then

(v,f)

the

of

function

(v E V),

(V*,g),

= vf

(v E V,

then f E

f:u ~ f u

is a l i n e a r

(U,V)

U).

(u E U),

isomorphism

is a dual

Conversely, where

of

pair

if

f u :V ~ &

(U,&)

onto

-is -

a

of

The

proof

consists

of a s i m p l e

application

of r e l e v a n t

definitions

as an e x e r c i s e .

If c o n v e n i e n t , as a t - s u b s p a c e U.

U

the b i l i n e a r

is a dual

fU = nat

is

is u s e d

convenient

with

defined

(~

(v,u ~)

be

precisely,

(U,&,V)

for

xf # 0

l.l.l &

=

is a pair

is e q u i v a l e n t

(or b r i e f l y

that

More

(~,~) vi)

(&,V)

subspace

(U,V)

4.

ring

the n o t a t i o n

vi ~)

that

the n o t a t i o n

the d i v i s i o n

space.

we m a y

space

if

(v,u)

over

vector

if it s a t i s f i e s :

v)

which

on a right

vector

(v,u)~ = (v,u~),

A bilinear

[7],

and

is a left

(ov,u) ,

V , V l , V 2 ~ V,

over

analogously

space

transformation

a bilinear

(Vl+V2,U)

where

spaces

of a right

analogously.

i)

iv)

and

of right

the

ii) iii)

such

on

and

defined

For 4,

transformations

as o p e r a t o r s

of

A dual

consider

V

we w i l l V*;

we

identify call

discussion

fU

u

fu' w h i c h

the n a t u r a l

is v a l i d

as a t - s u b s p a c e

and

of

imsse

by i n t e r c h a n g i n g U*.

amounts of

U

to c o n s i d e r i n g

in

the roles

V* of

U

and w r i t e U

and

V,

so

For a given of

a

in

U

(U,A,V)

(and

a

(v,bu) I.i.2 LEMMA. U,

if so denote

subrin$ o f

:

For a fixed b~u.

Then

b

and

=

(v,bu)

u, we o b t a i n Since

just

b~

V)

that

b

is an ad~oint

if

(2)

introduced,

a

has at most

~:a -- a*

one adjoint i n

is an isomorphism

of a

£(U,t~).

are adjoints

= (v,b'u)

(v,bu) =

u E U

in

the function

onto a subrin$ o f

If both

b

we say

(u E U, v E V).

the n o t a t i o n a*.

b E ~(U),

of

(va,u)

With

(va,u)

bu=

a E ~(V),

is an adjoint

it by

£(A~V)

PROOF.

and

(v,b~u)

is arbitrary,

of

a

in

U,

(v E V,

u E U).

for all

v ~ V

we have

then

(2) yields

which

b = b s, which

implies

proves

that

the uniqueness

of the adjoint. Suppose

that

s

has an adjoint.

Let

c~,~ E A, x,y E V;

then

for any

u E U,

we get

((ox+~y)a,u)

=

(ox+~y,a*u)

= o(x,a*u)

= o(×a,u)+T(ya,u) so that

(Ox+~y)a

Reversing the unique

= o(xa)

the roles

adjoint

Suppose

of

that

+ T(ya),

and

= a

of the adjoints,

a*

in

have

(~(×a)+T(ya),u)

is linear. we see

V, and that

a,c E 3~(V)

+ ~(y,a*u)

a*

adjoints

that

this

also

shows

that

a

is

is linear in

U.

Then

for any

v E V, u ~ U,

we obtain

(v,(a*+c*)u)

= (v,a*u+c*u) = (va,u)

which

implies

that

a*+c*

(v,(a*e*)u) which

implies

have

an adjoint

onto a subring on

U,

then

a

of

U

forms

£(U,g).

must

1.1.3 LEMMA.

For

i)

rank

(ab)


there exists a set e = el+e2+...+e iv)

the m u l t i p l i c a t i o n

is performed according to the usual row by column rule

0, e E ~u(V)

el,...,e n

For

a =

is an idempotent of

rank n

of o r t h o g o n a l idempotents in

if and only if

~2,u(V)

such that

n

Prove that every nonzero ideal of a transitive subsemigroup of

g(V)

is

itself transitive. v)

Prove that every nonzero ideal of a dense subring of

~(V)

is itself

dense. vi) Va

Show that an element

s

of

~(V)

is idempotent if and only if

elementwise fixed. vii)

For

a,b E g(V),

prove that

frank a - r a n k b l k,Ulp)~pVkp

such that

i < k,p < n, and

for

are linearly

e'U,

n

~n

xke = xke

form a basis of

(U',V')

(Xk,U i ) = 6k p P

v 1 ,...,v I n

the vectors

with

2 = k=l ~ p=l i [ik,~k~, (xgj )vj , we then obtain

T

b*U ~ b ' V * c

fl'''''fn

product

=

b'V* c

{~ij] E Ln(A ).

the multiplication

Vbc = Vc; similarly f.j

that

is generated

and

that

0 Hence

T

b*U = b * v = T

Recall

follows

2 (xfj)(v j=l

Consequently

and

bc = { b c

we

It

=

n c*U gj = i~=lUi ~ij

for

i _< j _< n

(x 6 v),

= dimVc

= n,

31 n

c*U ]vVC (xb)c = J~--I[ (xb) ~ n ~ i~__lUi Yij ) j n n n b*u ]vVb~ n c*U Vc ~ [X(m~=~lum ~mk ) k j (if~=lUi Yij)vj j=l [ k=l ~

~ub,U

n n ~,. [ ~ . ( Z v , Vb u.c~U k=l mK i= 1 K i

x

j=l

j=l

m=l m

'

)~..]~ lj

vVC j

x( ~ u b*c*U , Vbc x(bc) m= I m °mj)Vj =

(2)

where we have set • Vb

c'U,

[oij] = [~ij ] IV i nj By hypothesis

rank(bc)

= n, so that

] [~ij ] .

(2) together

with

(i) yields

e:bc ~ (b*c*U,[oij],Vbc) since

the linear

forms f. and the matrix [~ij ] are unique. J ~ Vb c*U~ and thus also Iv i uj j and we obtain

is invertihle

(b~)(c@)

, Vb

c*U~

] [Yij ],Vc)

(b*c*U,[oij],Vbc) Suppose

next

that

1 ~ i < n, let

the hypothesis

rank(bc)

dependent. Hence, n XlbC = i~=2oi(xibc) with

rank(bc)

x.1

< n

Then

implies

bc = 0

in

V

with

for some

o i.

in

For

(bc)@ = 0.

xib = vVbi "

XlbC,...,XnbC

we may suppose

i _< j _< n, let

Then for

Q, so that

the property

that the vectors

vb c*U v i uj = (xib)(c*hj) and

= (bc)~.

loss of generality,

c*U c*h. = u . J J

the property

< n.

be any vectors

without

[~ij ]

= (b*U,[~ij],Vb)(c*U,[Yij],Vc) = (b'U, [~ij ] IV i uj

For

Consequently

h.j

Then

are linearly

that

be any vectors

in

U

i < i,j < n, we obtain -_ = (xibc)h j

hence

Vb c*U

v I uj which

implies

n n = (XlbC)h j = [i=2 ~ o.(x.bc)]h. i i J = i~_2°i(xibc)hj

of the remaining (b@)(ce) 3.

= 0

rows and thus

which

completes

To show that

(b*u,[~ij],Vb)

Vb c*U~ [v i uj j

that the first row of the matrix

~

r Vb

c'U,

iv i uj

j

that

is one-to-one,

let

= (c*U,[Yij],Vc)

with

is a linear combination

is not invertible.

the proof

[~ij] = [Yij], Vb = Vc, which

n ~ Vb c*U~ = i=2 ~ o.~v. i i u.J )

0

b,c E Q \ O

the notation

by (i) evidently

Consequently

is a homomorphism. and suppose

as above.

implies

that

Then b = c.

that

b*U = c'U,

32 4.

To show that

e

~n A f.j = =If~lu~iji

Then for Since

is onto,

for

let

[~ij ]

is invertible,

are linearly

[~ij] E Ln(~),

n B xb = J=~l(xfj)vj

i < j < n;

i omorphism

isomorphism

is an equivalence of of

(U,&,V) (UI,&I,V ~)

onto onto

defined by = a

-i

ca

(c E £U(~,V))

ont_____o £U~(&

,V ), and

of

relation.

rank c = rank(c~(~,a) ).

39 We wish

to show that every isomorphism

of a semigroup

S

onto a semigroup

S ~,

for which $2,U(h,V) with

dim V >

isomorphism

several

important

(~,a)

of

dim V = I.

Let

of Section

O

8

/ P%~i'

/ = (v%,

define

,Iv;P ) i )'

functions

Then

(w,a)

a

and

b

i.

then

Since

of the general

S = $2,U(&,V)

case;

onto

E V, u i E U ~, v ki fi V I

and

S~

as Rees matrix

be as i n 5 . 6 .

as

at

~h__!e b__e_e-

semigroups

O

t-

Suppose

that

S ~ = ~ (Iu~,~&

and

.

,Iv~,p

J

)

with

dim V >

1, a n d

b_~

= u ici(Tw - i)

i C U'). (ui~

isomorphism

of

E V),

(U,~,V)

(ii) (12) onto

(U',~/,V ~)

with adjoint

= ~, of

the

first

the representation

Ow = 0 1

~

for

proves

2.

S

of

b(u~T)

implies

= (~w)d v ~

statement

consists

of

of nonzero

elements

a verification

of

the

= ~

b

0a = 0 I, a

# 0 I.

so that

It follows

~ = 6.

of

V

in the form

is single valued. that

We thus have

~

If

= B~

~v

=

and hence

~v

and

?v X

that

a

k~ _l]a maps

is defined

one sees that For any

a

k = ~.

But

is one-to-one.

= [(~d ~ -I- l)w-lw]d V

onto

i v~k ~ -~~ =

~v~~

(13)

V'.

in a way completely b

~-

is a one-to-one

analogous function

to that of of

U~

onto

a, so by a s~milar U.

o E ~, ~v k E V, we obtain by (ii),

[o(~vk)]a = [(~)vk]a = (o~)~dkv'k~ = (°~)[(v~)d~vk~ ] = (o~)[(~v)a] which verifies For any

is

(~vx)a = ( 6 v ) a # 0,

0 ~ ~ Wv~ E V ~, we obtain

/(~d k~--I l)w-lv

Now

false

V)

(vw)dxv~

argument,

for some semi-

then establish

in 5.5.

then also

which

for the treatment

(~v

The p r o o f

and

Further

{(w,a)

We will

and show that it is in general

= (~)dkv~

I$2, U (~,

unique,

(U',A',V').

(~v)a

b and ¢(w,a)

hypotheses

u i E U, v both

~,c,~,d,~

is a semilinear

PROOF,

is crucial

Phi = (v~ ,ui)

with

and l e t

U I

of an isomorphism

be an isomorphism

2, consider

-

S = ~ (Iu,~&

~ S' ~ gU,(h',V')

[4].

= $2,U ~(~ ' ,V ' ), fix vectors

$innin$

S

onto

of this result

The next result

1.5.9 THEOREM.

to

(U,h,V)

consequences

it is due to Gluskin

S'

$2,U,(£',V')

i, is the restriction

linear

for

c S c gU(&,V),

(4). ~vx E V

and

u~l E U ~, we obtain

by (7),

(ii),

and (12),

40

(7vk,b(u~T))w = (~vk,u ici(T~-l))w d

= (~) which verifies

3.

Let

(xia,b

-i

Xl,X2,X 3

a

is additive

be linearly

such that

and will do this in several

independent

(xi,gj) = 6ij

gj)1 = (xi,gj) ~ = 6ijw = ~ij

for for

vectors

in

1 ! i,j < 3. I < i,j < 3. -

for

a

/

,uiT)'

(5).

We will now show that

gl,g2,g3 E U

= (3~0)[(vk,u i)ci]wT

~ J ~T I u I x(vX~,ui ) = ((~w)dxvx~ , iT) ` = ((~vx)

V.

steps.

There exist

By (5), we have 3 If ~ o.(x.a) = 0, then

--

i=

1

z

i

i < k < 3, we have 3

3

-1 o k

and hence 4.

= Ok(Xka,b

xla,x2a,x3a Now let

x

-

gk) =

~ o.(xia,b i=l I

are linearly

and

y

-

lgk) = (i~__lOi(xia),b

igk) = 0

independent.

be linearly

independent

vectors

in

V.

The vectors

xa,ya,xa+ya are linearly dependent so by part 3 we have that the vectors -i -i -i xsa ,ysa ,(xa + y a ) a are linearly dependent. Since x and y are linearly -i independent, we must have (xa + y a ) a = ox+Ty for some o,T E 4, whence xa+ya Using

= (ox+Ty)a.

(14) and (5), we obtain for any

(14) u I E U ~,

(x,bul)w + (y,bul)~ = (xa,ul) I + (ya,u')' = (xa + y a , u l ) ' = ((Ox+~y)a,ul) I = (Ox+~y,bul)~ There exists yields

u E U

lw = o~.

such that

=

[O(x,bu I) + T ( y , b u l ) ] ~ .

(x,u) = i, (y,u) = 0; letting

Consequently

o = i

and analogously

(15)

u ~ = b-lu,

T = i.

Formula

(15)

(14) then

becomes (x+y)a for the case in xhich 5.

Again

let

x

z

are linearly

It follows easily

so that

xa + y a

vectors

x,y.

y

(x,y E V) are linearly

If

x = 0

but both nonzero,

independent

that if

linearly independent. xa+ya+

and

x,y E V.

are linearly dependent and

= xa+ya

x+y

Using

= (x+y)a,

y = 0, (16) is trivial. z

be a vector

(here we are using

(16) several

which a

let

independent.

~ 0, then both sets

za = x a + ( y + z ) a

Therefore

or

(16)

implies

V

the hypothesis x,y+ z

and

If

x

such that that x+y,z

and

y

y

dim V >

i).

are

times, we obtain

= [x+ (y+z)]a

is additive.

in

= [(x+y)+

that (16) is valid

sin = ( x + y ) a + z a for linearly dependent

41 By 5.5, (m,a) adjoint 6.

is a semilinear isomorphism of

(U,~,V)

onto

It remains to verify the last statement of the theorem.

[i,~,~] ~ 32,U(~,V )

and

with

= ~(od-i l)~-iv = od-l-i [(v

= [(ov~)a-l} [i,~,X]a

_i } [i,v,k]a = [(od -I_I)~-I( v

-l'Ui)~}wdkvk~ = od-l-i (p

_l,Ui)~vx}a

-i .~)(~)dkvk~

-

= od -I i d

~-

~

_i p'

-i

_li(C !li~)wdkv~

~,~

~(~,a)I~2,U(~,V)

i

= o(v',u' , )'(c _li~)~dkvkl

~

= (ov~)[~-li,(c~!liV)~dk,k~] which shows that

~-~i

= (~v~)@

= @

and completes the proof.

1.5.10 DEFINITION. Let S be a subsemigroup of gU(g,V) U ~ { i ,V i). An isomorphism ~ of S onto S i

and

semigroup of

by a semilinear isomorphism =

4'

For

ov I~ E V', using (13), (ii), (7), (6), we obtain

(ov~)(a-l[i,?,X]a)

~(~,a) Sl

(UI,~',V I)

b.

(~,a)

of

(U,g,V)

onto

S'

be s sub-

is said to be induced

(U',&',V ~)

if

This same definition is to be used for subrings of

£U(~,V)

and

£U,(8',V'). In particular 5.9 says that every isomorphism of is induced by a semilinear isomorphism.

$2,U(h,V)

onto

$2,U,(& ,V ~)

We will show that the conclusion is valid

under much less restrictive circumstances. 1.5.11LEMMA. 32,u(V )

If

is a subsemisroup of

gu(V)

containing

~2,u(V),

is the unique nonzero ideal contained in every nonzero ideal of

PROOF.

Write

52 = $2,u(V).

52 ~ S ~ ig(v)(32) S, and let

and thus

a C I\0.

ba E 32 N I. But then

S

For any

c E I

52

By 2.8, part i), the hypothesis is an ideal of

S.

By 3.3, there exists

b ~ 32

c C 32 , there exist

g,h C 32

and hence

Let such

I

S.

implies that

be a nonzero ideal of

ha # O; also

such that

c = g(ba)h

For semigroups of matrices,

1.5.12 THEOREM. Sl

Let

be a subsemigroup of

dim V > i, and let

~

by 4.7.

32 ~ I.

We are now able to prove the main result of this section, due to Gluskin

guise, and then by Gluskin

then

this was first proved by Halezov

[4].

[2] in a different

[3]. S

be a subsemigroup of

gUI(&I,V ')

containing

be an isomorphism of

S

gU(&,V)

containing

32,U,(~',V'), onto

S ~.

Then

$2,U(g,V),

suppose that ~

is induced by a

42 semilinear isomorphism

(o0,a) of

unique extension of

to s homomorphism of

PROOF. ideal of

~

~2,U~(f~,V~),

Letting

hypotheses of 5.9, so

~

i ~ IU

$2,u(A,V) S.

gu(A,V)

V~

into

is the unique nonzero

must map e

gU~(A~,V ~) (~)dxv~

V ~, and this representation is unique since such that

~

we see that

in the form

(vX,u i) = o ~ 0

so that

is the

Since the same kind of

a

~2,u(A,V)

satisfies the

is induced by a semilinear isomorphism

q0 be a homomorphism of

onto

and ~(u~,a) guI(A i ,V J).

into

it is clear that

e = ~I~2,u(A,V )

write an arbitrary element of V

(U',AI,V l)

gu(A,V)

contained in every nonzero ideal of

~2,uI(A~,VI).

Let

onto

First note that by 5.11, we have that

S

statement is valid for onto

(U,A,V)

(w,a).

which extends

~.

as we may since is one-to-one.

~v X = (~vx)[i,o-l,kl.

a

We maps

There exists

Hence for any

c ~ gU(A,V), we obtain [ ( ~ ) d k v ~ ]a-lca = (~vk>ca = [ (~v k) [i,(~-l,k]}ca

(~vx)a[a-~([i,o-~,X]c)a} = (Vv~)a{ [i,o-~,X]c}~

= t (Vv~) [i,c-~,~l}a(c~) = (~v~)a(c~) = { (w)d~v~J (c~) and thus

cop = a

-I

ca.

Consequently

same calculation with Since

cp

is the unique extension of

~

to a homomorphism of

it is also the unique extension of

Defining

£(&1,Vl).

contained in

~(w,a)

~(~,a)

~U(&,V), we see that

~(~,a)

~UI(&I,V I)

if

subsemigroup of

U~

~

gu(A,V)

containing

into

to such a homomorphism.

~

~U(&,V)

onto

to an isomorphism of

is contained in

~(~,a)

to s, the ~(0~,a)IS = ~'

by the same formula as on

SU(~,V),

preserves rank, and Ii~u(&,V )

are t-subspaces of

~2,U,(&,V)

then

Sl

~(&,V) is

~(~,a)I~U(~,V) into

is

[~UI(&~,VI).

V*, then every isomorphism of a

~2,UI(&,V )

onto a subsemigroup of

can be extended to an (additive and multiplicative)

~(~,V).

1.5.13 COROLLARY. be a sabring of

£(&,V)

to a homomorphism of U~

8UI(g,V)

containing

automorphism of

since ~

and

~

is an extension of S

e

c ~ S, shows that

is in fact an isomorphism of

on all of

For example, if

the unique extension of Furthermore,

let

extends

and for

~e know by 5.8 that

gU,(~,V)

~

~0

£UI(~J,VJ).

onto

q0 = ~(0~,a)" Since

instead of

gu~(AI,V~),

~

Let i

i

~Ut(~ ,V )

be an isomorphism of

~

be a subrin$ of i

containing ~

onto

5.12 and is thus a ring isomorphism of

£U(g,V) i

containing

~UI(~ ,V ), suppose that rD.~~. ~

Then

onto

~.

~

~U(g,V),

~

dim V > i, and

satisfies the conclusions of

43 The ring version restriction

of 5.12, with

on the dimension of

the rings

~l = ~u,(V) ' the result was established

isomorphism

every multiplicative cannot be dropped. ~(V') ~ 4'.

~'

as in 5.13 and no

[3]; for

the remarkable

~ = Su(V),

[2].

property

that every

of a ring in that class is automatically additive,

isomorphism For if

is a ring isomorphism.

The hypothesis

dim V = i, it is easy to see that

But there exist division

multiplicative

and

earlier by Dieudonn~

In 5.13 we have a class of rings with multiplieative

~

V, is due to Jacobson

isomorphisms

which

rings

(even fields)

are not additive.

i.e.,

dim V >

i

~(V) ~ 4; similarly

for which

For example,

there exist

let

4 = 4~

be

the field of real numbers and n be any odd integer, n # i; then the mapping n is a multiplicative automorphism of ~ 4 but is not additive. It follows

x ~ x

that the corresponding isomorphism which

automorphism

in turn implies

of

£(V)

that

is not induced by a semilinear

dim V >

1

cannot be dropped in 5.12 or

5.13. For

0 < ~ _< (dim V) +, let

1.5.14 COROLLARY. that

PROOF.

Then

If

~

Conversely and

~

i) ii) iii) iv)

v)

4

and

if

of

4'

g ,U(g,V)

is induced by a semilinear

(w,a)

¢(w,a)

preserves

is a semilinear

~ = 4 ' , then

be cardinal numbers

statements

onto

such

__if --and onl~ __if S = 4 ~

g4,,U,(4~,V~),

isomorphism

rank

(~,a)

we must have

isomorphism of

and

(4~,V ~)

then by

of

(U,4,V)

~ =

(U,4,V)

is the required

~(~,s),l~

4, U(g, V) For vector spaces (A,V)

1.5.15 COROLLARY. the following

and

g4,U (4'V) ~ ~s',U ;(g~'V~)

is an isomorphism

(U ,& ' ,V ' ), and since

(U ,4 ' ,V ' )

i

fl ~U(4,V).

(UJ,4J,V').

5.12 we know that

onto

isomorphism. with

dim V >

i,

are equivalent.

& ~ 4', dim V = dim V' (4,V) ~

(4~,V').

g (4,V) ~ g

4

~(4',V ~)

for some cardinal

numbers

Q,4 ~ >

i.

g(~,V) ~ g(a',V').

£(4,v) ~ £(a',v').

PROOF. one-to-one a

dim V >

i < 4,s' < (dim V) +.

an__~d (U,a,V) ~

onto

Let

g4,U (&'V) = ~4(&'V)

i) = ii).

Let

function mapping

to all of

V

w

be an isomorphism

a basis

B

of

V

of

4

onto

onto a bssis

by defining n

va = k~=l(~k~)(VkS)

n if

V = k~=l~kVk

(v k E B).

~ B~

and of

V ~.

s

be a Extend

44 Verification

that

a

is a semilinear isomorphism of

(~,V)

onto

( ~ , V ~)

is left

as an exercise. ii) = iii).

This follows

iii) = iv).

This follows from 5.12 with

iv) = v) and v) = i).

from 5.14. U = V*, U ~ = V ~*.

This follows from 5.13 with

This corollary says that under the hypothesis determines

(~,V)

the semigroups

and the ring

We now investigate when two semilinear (or ring) isomorphism. of

G

induced by

If

G

g, i.e.,

1.5.16 DEFINITION. (x E V)

(~,V).

V

Let

m

(&,V)

U

of

V*

then

a

PROOF.

Let

a

defined by

xm

= ?x

is a semilinear auto-

but most of the proof is old.

be a vector space.

If

is s multiplication.

i E IU

the hypothesis

dim V,

?, or simply a multiplication.

a

is any function $2,U(&,V)

Conversely,

for some

every multi-

£(V).

be as in the statement of the theorem.

and there exists

~2,u(V),

and

the inner automorphism

into itself which commutes with all elements of

t-subspace

g

g

~ ~ O, m~ = (~ _l,m )

~ l i c a t i o n commutes with all elements of

x = ~vl_

i, any of the following

induce the same semigroup

~ ~ &, the transformation

The next result is new,

1.5.17 THEOREM. mapping

in

isomorphisms

c

It is easy to verify that for

U ~ = V ~* .

the division ring

is a group, we denote by -I xc = g xg (x E G). g

For

and

~(~,V).

is called the m u l t i p l i c a t i o n induced by

m o r p h i s m of

dim V >

up to a semilinear isomorphism:

g (~,V), g(~,V),

U = V*

such that

Let

(vX,ui) = o # O.

0 ~ x E V. Since

Then

[i,o-l,k]

is

implies

[(xa,ui)o-l~-l]x = (xa,ui)o-lvk = x(a[i,o-l,~]) = x([i,o-l,k]a) = (~(vk,ui)o-lvk)a = (~vx)a = xa. It follows that

a

some scalar

we have

~x

there exists

maps

b E ~2,u(V)

x

into the subspace of

xa = ?x x.

If

such that

x

and

xb = y.

V y

generated by

x, and thus for

are nonzero vectors,

by 2.10

Using the hypothesis, we obtain

~yy = ya = (xb)a = x(ba) = x(ab) = (xa)b = (~xX)b = ~x(Xb) = VxY and hence x # 0. quently

~x

Let

=

o

a = m

Conversely,

~y.

But then

=

~x

be the zero function on

is a constant and we have V.

Then

if

b ~ £(V),

m b = bm

.

then for every

x E V, we have

= (~x)b = ~(xb) = ( x b ) m

xa

=

~x

0 = (0a)o = (Oo)a = 0a.

•

x(m b) = ( x m ) b and thus

~

= x(bm~)

for all Conse-

45 If C(S)

S

is a semigroup

[a ~ S I xa = ax 1.5.18 LEMMA. PROOF.

(or ring), by

for all

For

C(S)

we denote

the center of

S, i.e.,

x E S} .

~ ~ &, m

is linear if and only if

~ E C(&).

Exercise.

1.5.19 COROLLARY. PROOF.

If

a E £U(g,V),

C(£U(g,V))

=[m

IV E C(~)} .

a ~ C(£U(~,V)) , then

by 5.18 we also have

by 5.18, we have

m~ E £(g,V),

£(~,V).

for any

Further,

a = m

that

for some

~ E C(~).

and by 5.17, m

? E ~

by 5.17,

Conversely,

let

and since

~ E C(A).

commutes with all elements

Then

of

x E V, f E V*, we obtain

x(m~f) = (xmv)f = (vx)f = v(xf) = (xf)v = x(fv) which proves

that

m*f = f~

again a multiplication). and therefore

m

( ~ , V ~)

If

and of

a semilinesr

Consequently

(~,a)

onto

~ E & . PROOF.

( ~ , a ~)

into

of

Let

into

Then

that

m v E £U(&,V)

transformations

respectivel~,

then

of

(~0o0 ,as )

(&,V) is

and

( ~ , a ~)

The equation

a~a -I

-i

~(w,a) = ~(w~,a ~)

ca = a'-ica '

to

isomorphisms

if and only if

of

aI = m a

for

to

to

as

a~a -I = m

a ~ = m a.

is equivalent

(c E ~U(&,V))

commutes with all elements

is equivalent

be semilinear

I_~n suc____~h~ case, w ~ = ¢ _i w.

in turn can be w r i t t e n

equivalent

implies

is

(g~,VH).

~(~,a) = ~(~i,al)

(a'a-l)c = c(a' a -1) i.e.,

which

are semilinear

(&~,V~),

(A,V)

(~,a)

(UI,&I,VI).

a which

and

( ~ , V ~)

transformation

1.5.21 COROLLARY.

some

m*U c-- U

of a multiplication

Exercise.

PROOF.

(U,&,V)

the conjugate

E C(£U(&,V)).

1.5.20 LEMMA. into

(in particular,

for some

(c E £U(&,V)), of

~U(&,V).

~ E A-;

The last statement

(17)

Hence by 5.17,

the last expression

of the corollary

formula

(17)

is evidently

follows

from 5.20.

1.5.22 EXERCISES. A ring

(~,+,.)

is said to have unique addition

if for any ring

defined on the same set and under the same multiplication, This concept was introduced i)

by R.E. Johnson

Show that a ring

every isomorphism

of

~t

that

(~,-) + = ~.

[i].

has unique addition onto

we have

is additive.

if and only if for any ring

~,

46 ii)

A ring

~

all of whose elements are idempotent is called a Boolean rin$.

Show that a Boolean ring has characteristic

2, is commutative~

and has unique

addition. iii)

Show that the ring

Z

of integers has only one automorphism,

has an infinite number of automorphisms.

Hence

Z

wheras

~Z

is not a ring with unique ad-

dition. For any ring iv)

Let

~

~, by

3+

denote the additive group of

be a ring and

F

3.

be a prime field (i.e., F

is either the field

of rational numbers or the ring of residue classes of integers Prove that if ~ 2 = 0.

not additive F+

[F0~~ ~F,

then

~ ~ F, and if

Give an example of a prime field (and hence

F

mod a

~ + ~ F +, then either F

prime).

~ ~ F

w i t h an a u t o m o r p h i s m of

does not have unique addition)

or

~F

which is

and an automorphism of

which is not multiplicative. v)

Prove that if

~

is a ring for which

3 + ~ Q+/Z +

rational numbers over the additive group of integers), example of a semigroup vi) g(V)

S

Show that for a vector space

and any nonzero ideal of An alsebra

over a field (A,+,.)

4

then

with zero and with the property

(g,*,A,+,.)

V

~(V)

with

dim V >

(the additive group of ~ 2 = 0. S ~ ~0~

with scalar m u l t i p l i c a t i o n +

i, any nonzero ideal of

*

and addition

and m u l t i p l i c a t i o n

~(ab) = (~a)b = a(~b)

etc.

+, and a ring

(~ E 4, a,b E A)

dim V >

Let i.

(U,4,V) Show that

A.

be a pair of dual vector spaces where ~U(4,V)

x(~a) = ~(xa)

algebra i s o m o r p h i s m of inner a u t o m o r p h i s m of

(U,4,V) ~U(4,V) £(4,V),

4

can be made into an algebra over

is a field and 4

by defining

(x E V, ~ ~ 4, a E £U(4)).

prove that every algebra isomorphism of linear isomorphism of

The concepts of

for an algebra refer to tbe same notion for both the

vector space and the ring structure of vii)

(4,*,A,+)

., and satisfying

where we have denoted both "multiplications ~' by juxtaposition. homomorphism,

~.

have isomorphic a u t o m o r p h i s m groups.

is a system consisting of a vector space

with the same addition

subalgebra,

Give an for no ring

onto onto

~U(g,V)

(UI,4,V~). £U~(4,V )

onto

~UI(4,V~ )

Deduce that for

is induced by a V = V l, every

can be extended to an algebra

and that every algebra a u t o m o r p h i s m of

£(&,V)

is

inner. viii)

Prove that if

4

is a field,

leaves the elements of the center of

then every a u t o m o r p h i s m of ~(&,V)

£(g,V)

which

fixed is an inner automorphism.

47 ix)

Show by an example

is0

have a unique the form

addition.

that e subring

Such an example over a 2-element

a21 0

0:

all 0

0

~

a32 a32 0 is an automorphism

In fact,

need not

of all matrices

of

show that the mapping

~

a21

0

a31+alla21

a32

but is not additive.

Find another

addition

for

~I~

which

it a ring. x)

State

striction

and prove

xi)

Show

that if

and

Chapter

[4], Jacobson

[1],[2],

2, Sections

([7], Chapter

Mihalev

Gluskin

We will discuss

and Lam

[I], R~dei

ii),

/i], [2],

[1], (~2],

IX, Section [l], Johnson

If], Mihalev [i], Rickart

12), and

[i], [1],[2],

AUTOMORPHISMS groups

in the preceding

we write

all functions dim V >

auto-

Throughout

of functions

on a set

on the right. i,

auEomorphisms

automorphisms

of semilinear

section.

Multiplication

space with

is the group of all linear

Johnson

Martindale

of certain

is the group of all semilinear A

Jacobson

and Steinfeld

notation.

is a fixed vector

/I],[2],

IV, Section

/2],

[5],[6].

only a few properties

is taken to be their composition;

of

see

6), Dieudonn~

[I], Fajans

12),([6], Chapter

OF SEMILINEAR

encountered

gidelheit

[i], R.E.

[1],[2],

[i], Wolfson

we fix the following

(&,V)

Halezov

transformations,

II, Section

ii), ([7], Chepter

III,IV),

3, Section

GROUPS

we have essentially

this section,

IX, Section

[3], Mackey

/i], Morita

1.6

the re-

with all idempotents

and semilinear

([i], Chapters

12), Jodeit

[i], Stephenson

and commutes

([i], Chapter

[2],[6],[7],

[i], Kaplansky

and Satalova

morphisms

on isomorphisms 4), Behrens

Baer

5.14 and 5.15 without

must be a multiplication.

7,8 and Chapter

Y

[3], Rjabuhin

is additive a

[3], ([6], Chapter

IV, Section

Kiokemeister

in 5.17

V, Section

and for related material

of 5.12,

V.

i, then

information

([1], Chapter

Gewirtzman

of

a

dim V >

For further

Gluskin

the ring version

on the dimension

~2,U(&,V),

Baer

of

field.

addition

~

Jail001 Lall 0!]

a31 a32

makes

of a ring with unique

is given by the ring

of

of V

V, (invertible

linear

transformations), M

I

is the group of all multiplications is the group of all semilinear

(m

= (e

,m ) with ? inner automorphisms (i.e.,

W E ~ ),

48 £ = £(A,v). Furthermore,

for

S

a semigroup

(or ring),

let

~(S)

be the group of all automorphisms

~(S)

be the group of all inner automorphisms

where an inner automorphism

of

S, of

S,

of a semigroup or a ring is defined by g -I (x ~ S), as in the case of groups, with the provision that g exists,

must have an identity If set

H

e

element and

g

is a subset of a group

{g E G I gh = hg

for all

must have an inverse

relative

G, then the centralizer

h E H}.

For subsets

H

of

and

H

K

-i = g xg g i.e., S

x¢

to it. in

of

G

is the

G, let

HK = [hk lh E H, k E K]. PROPOSITION.

1.6.1

Further,

A

a normal

subsrou~

is the centralizer

PROOF.

o_f~ ~ ,

~

onto

subgroup

6(£)

o_~f ~

PROOF. For any

with kernel

and of

By 5.8,

c E ~

implies

5.21, M

and

thus

that

~

[

for any

M

and is a normal

M ~ A = im

maps

and maps

~

into

of

~

subgroup

I ~ ~ C(a-)}

onto

of

~,

M.

I

i__~s

= C(M).

~.

~

maps

and by 5.13,

= ~(w,a)(~,b) M

Hence

¢

M

maps

is a normal

~

onto

6(£).

= (ab) -i c (ab) = c¢(w~,ab)

and hence

~

is a homomorphism.

is s normal subgroup of

m

I

$(~).

we have

Hence

7 E g , we have

to show that

onto

= b -I (a-lca)b

(a-lca){(w,b)

~(w,a)~(~,b)

((~,a) E >~), is a homomorphism

I/M ~ $(£).

6(£),

(~,a),(w,b) E ~ , =

I

~ / M ~ ~(~),

is also a normal subgroup of

It remains

i_nn ~

is an isomorphism

~:(w,a) ~ {(w,a)

M,

and

is the kernel of

Since

M

I = MA, a n d

Th__.~emappin $

c~(~,a){(~,b) which

of

~ ~ m -i

Exercise.

1.6.2 THEOREM. of

The mapping

= (¢ _l,m ), it follows I

onto

and the kernel of $(~).

Hence

let

~

~

and

that

M c

restricted

(¢ ,a) E I.

By

~ / M ~ 6(£). I

and to

I.

For any

o E g, x E V, we obtain (ox)(m a) = (7ox)a = ~-l(~g)~(xa) = c[(~x)a] and hence

m a

is linear.

= o~(xa)

= o[[(~e

_l)x]a}

= o[(xmv)a ] = o[x(mva)]

This

together with 5.20 yields

C(~v,a) = ~m a = e m a E a(~). 7 Finally, ~Z~,a~

is closed under addition

~

is also s division

dim V ~ = i

must have the same property,

and

K ~ = ~2,U ~(V ~) = ~2,V~,(V ~).

is an r-maximal

have

~, ~

isomorphism

the image of

Kl

~ E g-, we obtain

l i ~ I ~, ~ E ~] that

so

% E IV~

~°(I~,~I~-,Iv~;P I)

is a semilinear

that

and

and the functions

plicit hypothesis

Consequently

If for some

By the isomorphism

1.5.6 applies

ring,

But then

Suppose next that

j E It

Consequently

We let

is a division

to have only one element.

If

S # V, then there exists

Ss = 0. that

Then for any

a'V* ~ S ~

a nonzero

s E S, f ~ V*, we

and thus

0 # a E ~

(V). S~

Consequently, that

if

L = ~v,(V) iv) = i).

Further e E ~ which

let

A

by 1.6. is minimal

~r,~(L)

(U,V)

S = V

which

implies

= ~(V) = ~. Let

~

satisfying

be a minimal Hence

~I

iv) be a right ideal of a simple ring

right ideal of

= (e~)~ ~ c ~

that

~ = ~(V)

~

~.

= A

in view of its minimality

ring, and we may suppose 1.3.4, we have

= 0, then by (i) above we obtain

in

= ~u~(V )

for some subspace

Then

A = e~

so that ~.

Thus

where U

A~

U~ of

for some idempotent

is a right ideal of

~

~,

is a simple atomic

is a t-subspace

U~

~I.

of

V*.

By

and by 1.3.8, we have that

is a dual pair. The following argument

and notation

dual of 1.3.4 and (i) above, we obtain

is relative

to the dual pair

that if s subspace

S

of

V

(U,V). has

(V) = 0, then S = V. Let 0 # f E V*. Then S = Iv E V I vf = 0} S• subspace of V, and hence by the preceding statement, we must have ~ Consequently

By the

the property

is a proper (V) # 0. S±

88

s~ = [ u ~

ols~ = 0

Let

0 ~ g E S ~, then for every

and

g

sE

for all

v E V

are linearly independent,

s] ¢ O.

we have:

if

vg # 0

according to 1.2.3 applied to the dual pair

f = g~

for some

r ~ g-

which implies that

vf = O, then

then there exists

so that

f E U.

v ~ V

vg = 0.

such that

If

vf = 0

(V*,V), a contradiction.

Therefore

U = V*

and thus

f and Thus

U = U~

~ = ~.

The equivalence of ii) and iv) in the next corollary is due to W o l f s o n [i]; the rest is new. II.3.6 COROLLARY.

The following conditions on a ring

i)

~

is a primitive ring with an r-maximal socle.

ii)

~

is isomorphic

~

are equivalent.

to a ring of linear transformations on a left vector space

containing all transformations of finite rank. iii)

~I~

iv)

~

is a dense extension of an r-maximal completely 0-simple semigroup. is a primitive ring with a nonzero socle

a left ideal PROOF.

L

implies

i) = ii).

~u(V ) ~ ~i ~ £ u ( V ) But then

We know by 1.25 that

for some dual pair

~u(V) = ~(V)

ii) = iii).

and thus

We may take

is a dense extension of "ii) = iii)" in 3.5 that iii) = iv). L

a ~ ~r,N(L) that

~

and hence

~

ba # 0

and thus

for some ring

Since

~

~i

for

satisfying

is r-maximal, so is

for a left vector space

by 1.25.

~u(V).

V.

Hence

It was shown in the proof of

is a primitive ring with a nonzero socle ~r,N(L) = 0.

for some

Further let

b ~ L.

xa ¢ 0

~r,~(LA~)

and = 0.

0 ~ a E ~,

~.

Let

then

It follows from 1.25 and 1.3.6

and thus there exists

x = bach, we have

a ~ ~r,~(Ln~),

~r,~(L) = 0

is an r-maximal completely O-simple semigroup.

such that

is a regular ring,

Hence for

in which

~ £(V).

$(V) c ~ c ~(V)

$2,v,(V)

~

N ~ ~

(U,V).

~(V) ~ ~

~2,v,(V)

As before,

be a left ideal of

~

L ~ ~.

c E ~

x E L n ~

such that

ba = bacba.

which shows that

By 3.5 we must have

L N ~ = ~

so that

L ~ ~. iv) = i). ~r,~(L) = 0. 0 ~ a E ~, so

Let

L

As above

then

ab # 0

be a left ideal of the socle ~

is regular~ for some

Lab ~ 0, w h i c h implies that

hypothesis implies

that

L ~ ~.

and thus

b E ~ La # 0.

L

~

of

~

and suppose

is also a left ideal of

by 1.25 and 1.3.3. Consequently

Now 3.5 implies that

But then

~r,~(L) = 0 ~

that ~.

If

0 ~ ab ~

and the

is an r-maximal simple

ring.

The next result characterizes rings isomorphic space

V; other c h a r a c t e r i z a t i o n s

to

can be found in Leptin

£(V)

for some left vector

([2], Part I) and W o l f s o n [i].

89 II.3.7 COROLLARY.

The followin$ conditions on a ring

~

are equivalent.

i)

~

is a maximal primitive ring with an r-maximal socle.

ii)

~

is isomorphic

to the ring of all linear transformations on a left vector

space. iii)

~

is a maximal dense extension of an r-maximal completely O-simple semi-

group. is a semiprime ring having a minimal

iv)

right ideal

and is m a x i m a l among the semiprime rings

~r,~(A) = 0 ideal with

~'

A

such that

having

A

as a right

~ r , ~ ( A ) = 0. The equivalence of i), ii) and iii) is an immediate consequence of 1.27

PROOF. and 3.6.

i) = iv). such that

By 1.25,

~

~r,~(A) = O.

and having ideal of

A

is a semiprime ring and has a m i n i m a l right ideal

Let

be a semiprime ring containing

as a (automatically minimal)

~.

By 1.6, we have

e,e ~ E A, f E B.

Then

right ideal.

A = e~ = et~ I

A = ~

and 2.8, by 2.5 there exist y E ~,

~i

~ e~ ~ = e 1 ~ a,b E ~

and

= A, so that

such that

Let

B = ~

~

B

A

as a subring

be a minimal right

for some idempotents A = e~ j.

e = ab, f = be.

In view of 1.25 If

x E B

and

then xy = (fx)y = (baba)xy = b(ab)axy = fbe(axy) E fb ~e/~~ = fbA c f~ = B

which proves that right ideal of ~

of

~.

B

~.

is a right ideal of Consequently

But then

the socle

~

of

Since each minimal right ideal of

it follows that their sum r-maximality of

~

~, we have

Now the m a x i m a l i t y of iv) = ii).

~

Again

~u(V)

~

~ = ~J

since

~

~

where

are precisely

Ri = ~ [ i , ~ , X ]

must also be a minimal

is contained in the socle

~

and thus also of

satisfies

as a primitive ring with socle ~ ~ ~l

B

is a minimal right ideal of

is a right ideal of

~u(V) ~ ~' ~ £u(V)

dual vector spaces by 1.25 and 1.2.8. ideals of

~

~.

the conditions in 1.25. ~ = ~

implies

for some pair

and

~(V)

coincide. = 0

semiprime,

vxa = 0

Since

then

for all

the m a x i m a l i t y of

X E IV • ~,

£(V)

Ri

[i,~,~]a = 0

(vD,ui) J O, so that

of

I V ~ ~, X E I v ] .

and we have seen that

a ~ ~r,£(V)(Ri), such that

R i.

(U,V)

the sets

Thus the hypothesis on

for some

~ = ~.

We have seen in 1.3.11 that the minimal right

It follows easily that the sets of all minimal right ideals of the rings

~r,~(Ri)

~, By the

~

implies

satisfies

But then

~'

for all

a = 0 that

~ ~ g, ~ ~ I V . which

for

which shows that

3' = ~(V)

is semiprime and

the conditions

is a minimal right ideal of

(v ,ui)w(v a ) = O

we conclude

that

~u(V), ~

in 1.25, ~(V).

There exists ~ 9 0

it is

If ~ ~ Iv

yields

~r,~(v)(Ri) = O.

w h i c h finally yields

By

~ ~ ~(V).

90 The implication Wedderburn

"i) = ii)" in the next corollary

(or Wedderburn-Artiu)

theorem,

these rings was given by Gluskin [4],([7], Chapter

IV, Section

11.3.8 COROLLARY.

[i].

is known as the Second

A multiplicative

characterization

For further characterizations,

16) and Wolfson

The following

[ii.

conditions

~

are equivalent.

~

is a simple ring satisfying

ii)

~

is isomorphic

to a full matrix

iii)

~

is isomorphic

to the ring of all linear transformations

iv)

vector

d.c.c,

on a ring

i)

dimensional

of

see Jacobson

for right ideals.

ring

&n

over a division

ring

&.

on a finite

space.

~

is a prime

(respectively

~

is a simple

primitive)

ring satisfying

d.c.c,

for right

ideals. v) 0-simple

PROOF. minimal

ring a n d

~

is a maximal

dense extension

of a completely

semigroup. i) = v).

The second part of the hypothesis

right ideal, which

for some dual pair by 1.3.14 implies

that

But then 1.7.10 yields which is a completely extension

But then

dim V < ~. that

By 1.27,

which implies

dim V < ~.

that

d.c.c,

Since

ii) ~

We deduce

vector iii).

space

that

~ ~ ~u(V)

and thus

dense extension ~D~

has a

for right ideals which

= ~(V)

Consequently

~

~ ~ £(V). of

is a maximal

implies

that

is also simple, we must have

Consequently

for right ideals by 1.3.4.

iv) = iii).

gu(V)

is a maximal

~

~2,v,(V) dense

semigroup.

ideals and is both prime and primitive

dimensional

d.c.c,

the second part of the hypothesis

(U,V).

that

by 2.8 implies

satisfies

Consequently

semigroup.

0-simple

for some dual pair

satisfies

~u(V)

[k~(V) = g(V)

O-simple

of a completely

v) = iv).

together with simplicity

(U,V).

implies

nat U = V* Hence

~

~ ~ £u(V)

~u(V) = £u(V)

by 1o3.13 so that

£(V)

satisfies

d.c.c,

for right

~ ~ ~(V)

for a finite

by 1.25,

from 1.25 and 1.3.14

that

V.

This is a well-known

fact proved in elementary

texts on linear

algebra. iii) = i). Further

~(V)

fies d.c.c, Again

First note that

= gv,(V)

where

~(V)

is simple by 1.3.6 since

dim V < ~

so that 1.3.4 implies

~(V) = ~(V). that

~(V)

satis-

for right ideals. the hypothesis

corresponding

hypothesis

on right

ideals in i) and iv) can be substituted

on left ideals,

be taken on a left or a right vector

and in iii) the linear

space.

by the

transformations

can

91 11.3.9 EXERCISES. i)

Let

ideal of

~

~

be a primitive ring with a nonzero socle

~.

Show that a right

which contains s m i n i m a l left ideal must contain

A nonempty subset

A

for some n o n e m p t y subset

of a ring B

of

~;

~

~.

is called s left annihilator if

A = ~%,~(B)

a risht annihilator is defined dually.

It is

clear that a left (right) annihilator is a left (right) ideal. ii) ~(V)

Let

V

he a vector space.

coincide with right

Show that the principal right

(left) annihilators of

dimensional if and only if every right

£(V).

(left) ideal of

(left) ideals of

Deduce that £(V)

V

is a right

is finite (left)

annihilator. iii)

Let

~

be a regular ring with identity.

right (left) ideals of ~(V)

~

a sum of two right

is a principal

right

(left) annihilators

For the first statement, if A and B 2 e = e and b, respectively, consider

Show that s sum of two principsl

(left) ideal of

is s right

~.

Deduce that in

(left) annihilator.

(Hint:

are principal right ideals generated by e + (l-e)bx(l-e)

where

(l-e)h

= (l-e)bx(l-e)b.) iv)

Show that in any ring the intersection of any set of right

annihilators is a right

(left) annihilator.

of any set of principal right v)

Let

= ~(V).

V

Deduce

that in

(left) ideals is a principal

be a finite dimensional vector space, U

~(V)

(left) the intersection

right (left)

ideal.

be a subspace of

V*

and

Show that ~u(V) = ~r,$( [c ~ ~ I c*U = 0}).

vi)

Give an example of a ring

~

and two principal right ideals of

~

whose

~

whose

intersection is not a principal right ideal. vii)

Give an example of a ring

and two principal right ideals of

sum is not a principal right ideal.

II~4

SEMIPRIME

RINGS

We are interested here in semiprime rings with a nonzero socle. us a general result,

This will give

several special cases of which will be treated in greater

detail in the two succeeding sections.

Some new notions will come in handy

(recall

1.15). 11.4.1 DEFINITION. every

i ~ I, we have

Ai, i E I, if =

~ Ai . iEl

~ =

A family

~AiJiE I

A. • ~ A. = 0. I j#i j

~ A. iEl l

and the family

of subrings of

A ring

~

~Ai}i61

~

is independent

if for

is a direct sum of its subrings is independent,

to be denoted by

92

11.4.2

DEFINITION.

subsemigroups denoted

by

Such

A semigroup

S , ~ ~ A,

S =

~ ~ o@A

a "sum"

S

if

S =

as follows.

zeroes

0~.

Let

Let

S =

with

zero

and

S S~ = S

is an orthogonal ~ S~ = 0

sum of its

if

~ # ~,

to be

•

of semigroups

proceed

S

U S

may be termed

~S~c~A_

[ U

(S

'internal".

Externally,

be a family of pairwise

\03]

U 0, where

0

disjoint

is an extra

we may

semigroups

symbol,

with

with multi-

~A plication

a*b

in all other groups

= ab

a,b ~ S for some ~ ~ A and ab # 0 , and ct c~ In this case also we say that S is an ortho$onal

cases.

S c~, ~ 6

theorem,

if

A, with

see D i e u d o n n @

the same notation. [3] and J a c o b s o n

For

the origin

([7], C h a p t e r

a*b

= 0

su.__mmof semi-

of a part of the next

IV, Section

3),

the rest is

new. II.4.3 minimal

THEOREM.

risht

Let

ideals

~

be a semiprime

assumed R

=

U

nonempty. Ri,

~

=

be the socles

of

~lR

i T j i ~k' "''' c o n t r a d i c t i n g the hypothesis.

must be finite

The proof of the s u f f i c i e n c y is left as an exercise. II.6.5 EXERCISES. i) for

Prove that each of the following conditions on a ring

~

~

is sufficient

to be a division ring. a)

[F~

b)

~

is a primitive regular semigroup.

c)

For every

is a regular ring with only one nonzero idempotent. 0 # e ~ ~,

there exists a unique

x ~ ~

such that

= axa.

(These conditions are trivially necessary.) ii)

Let

ideal of

~ =

~ ~ , w h e r e each ~ ~A ~ is a direct sum of some ~

~ a)

A

b)

each

is a simple atomic ring. and conversely.

is finite if and only if ~

is isomorphic

only if every minimal

~

Show that every

Deduce that

satisfies d.c,c,

for two-sided ideals,

to a matrix ring over a division ring if and

two-sided ideal of

II.7

~

satisfies d.c.c,

for right ideals.

ISOMORPHISMS

We consider first isomorphisms of some of the rings we have encountered in the two preceding sections.

After that we give a construction of all isomorphisms of

two Rees m a t r i x rings. 11.7.1 LEMMA. >emigroup socles of

Let

~

and

~

and

~

w h i c h is a division ring.

~

and = of

~

~

G

~ ~

~

and

~

of

~/ =

~

~

onto

onto

AI

and

~

onto

~ ~/t

onto

~.

= ~I~ ' we have an i s o m o r p h i s m

~

~ ~ = ~~ of

~i

and

~

b e the

has no ideal can be uniquely

By 6.1, we have

is a simple atomic ring.

~ and ~ ~ ~ It is easy to see that

such that

~

~.

@ ~ /, w h e r e

~IEA~ respectively. A

~

be an isomorphism of

~//, where each

aEA ~ % and ~11,

function ~

Let

and assume that

Then every isomorphism o f

extended to a ring isomorphism o f PROOF.

b~e semiprime atomic rings, ~

respectively,

~

onto

~ aEA ~

Hence

are the semigronp socles ~

for all ~i~ .

~ =

induces a one-to-one ~ E A.

Letting

106 i ,V )I , where ~--~U (h(~,V) and r~~ ~--:---~U,(£ i c~ (y in view of the hypothesis on ~. We know from 1.5.12 that every isomorphism @

By 2.8, we have

$2~U(£,V)

onto

isomorphism

@

restriction

of

SU(&,V)

onto

~'

.

~2,UJ(h',V'), of

~U(h,V)

~

to

where

onto

SU(L,V)

that

Since

'~a

admits

~ =

~ ~

r = r I + . . . + r n,

hence

of

~

extended

Using 1.5.12 again,

is the unique extension

a uniqu~ ~xtension

the function +

morphism

l, can be uniquely

•'u'(A''V')"

..

+

,~:r ~ rl~a I where

dim V >

of

@

of

to an

we see that the

to an isomorphism

of

SU j(A',v'),

We deduce onto

~

~

onto

~

to an isomorphism

defined

~'

Let

rn~ n

At each step,

~

c~

-

.

extension

of

by

r.i % '~'-i' is a unique extension

is the unique

Ii.7.2 LEMMA.

~

of

~

the extension

of each

isomorphigms

to an isomorphism

be an isomorphism

~c

of a rin$

of

[~

~

to an isoare unique,

onto

~'

onto a rin~

~'

Then

the function

e-:(~,o) ~x = [ k ( x e - l ) ] G ,

~vhere

onto

~(.~) PROOF.

-

(k,~)

((>..,0)

x0 = [(x6-l)p]G

w ~ = .~, r r6 ()-,0) ( ~ ( ~ ) and

(x ~ ~ ' ) ~

such t h a t

(r t ~).

For

x,y Q ~',

[(xy)

= {.),l(xy)e-£]}e

= [~[(x+y)6-1]]6 =

so that

~

x(ly) = x[k(ye-t)]6 =

which proves

that (~$)x

and similarly one-to-one From

=

},r x

=

and

x ( ~',

Ikr(X@-l)}@

= kx+Xy,

p-

is a right

translation;

further

= [ (x6-l) [;, ( y e -i ) ] ; e

= [(×e-1)p]ey

If also

that

= (x-6)y ,

(~2,~) ~ ~(~),

= {),[(x6-1+y6-1)]e

analogously

lL[~(xe-1)]e]

and onto follows r ( ~

= lX(xe-1)]ey

{.[(xe-i)oj(ye-1)]e

(~,p) ~ fZ(~').

o__[f N(~)

we o b t a i n

[k(xe-1)J6+[),(ye-i)]e

is a left translation,

is an i s o m o r p h i s m

= {k[(xe-l)(y6-£)]]6

= [[x(,,,:e-l)](ye-l).~e ~(x+y)

~ ~(~)),

~

then

= X-6x

is a homomorphism.

from the fact

that

@

we obtain

= [r(xs-l)j6 = (r@)x = krsX

shares

That

@

is also

these properties.

i07

so that

--Xr= Xr~, and analogously

II.7.3 LEMMA.

Let

~

Pr = PrG --

which proves

be an essential extension of a ring

be a maximal essential extension of a ring onto

It

more~ X

I ~.

~

onto

~

if and onl~ if

~

~r ~ =

X

~r~"

I, ~(I) = 0, 3 ~

Then ever~ isomorphism

ca_nn.be uniquely . . extended . to an isomorphism maps

the formula

__°f ~

into

~

of

~.

1

Further-

is a maximal essential extension of

I. PROOF.

Let

8

be an isomorphism of

the ring analogue of 1.7.5 implies that ~(I), that all

and ~

T' = 7(3t:I l)

Hence

onto

X = T~Tt-I

~(I)

It

m = T(3:I)

is an isomorphism of

is an isomorphism of

x E I.

I

onto

~(I ~) = 0

and hence

is an isomorphism of

3'

~(I')

Then

onto

~(I').

with the property

is an isomorphism of

~

into

3

into

Now 7.2 asserts

~t

~x ~ = nx8

for

such that for all

x ~ I, we have x~ so that

X

extends

Let also for any

~

r ~ 3

x0

~x@ T ~.

be an extension of and

6

to an isomorphism of

~

into

3 ~.

Then

x ~ I, we obtain

(r~)(x@) = (r~)(xw) = (rx)~ = (rx)x = (rx)(x~) = (rx)(x~) and similarly

(xS)(r~) = (x~)(rA).

r ~ - rk { ~ t ( I t ) .

Bun

~,(I')

~,Cl

t) e ~' = ~C~'} = o

~ t ( I t) = 0.

x ~ I

is an ideal of

which together with the hypothesis implies

Since

that

Consequently

3'

is arbitrary,

~'

it follows that

satisfying

is an essential extension of

r~ = rx

and thus

~ = X

since

It

r ~ ~

is

arbitrary. The necessity in the last statement of the lemma is obvious; from the fact that

~X

is an essential extension if

sufficiency follows

I'.

The next result is new. II.7.4 THEOREM. socle

~

Let

~

be a semiprime tins essential extension of its nonzero

and assume that

3

has no ideal which is a division ring.

ring which morphism

is a maximal essential extension of its socle ~

of the semigroup socle

~

of

~

onto

can b e uniquely extended to a rin$ i s o m o r p h i s m X

~

~

onto

PROOF.

Let

3 ~ ~

if and only if

Since

According ~.

~t = C ( ~ ) ,

to 7.1, ~

By 7.3, ~

R

Hence

the semigroup socle o__[f 3

onto Rt

3 t

admits a unique extension

Rt

be a

into

3 ~.

~

of

~t

Furthermore, ~.

It follows from 5.9 that

is a primitive regular ideal of

by 6.1 we conclude that

can be uniquely extended

Let

Then every iso-

is a maximal essential extension of

be an isomorphism of

is a primitive regular semigroup. ~t

3

X

~.

~t

is a semiprime atomic ring.

to a ring isomorphism

to an isomorphism

X

of

3

0

of

into

~ ~t

onto

108

Consequently If

~

~=

X

provides an extension of

is another such extension of ~'.

But then

=

~I~

The last statement

8

~ = C(R)

• = X

and

if every m u l t i p l i c a t i v e

~/ = ~(~')

exercise

i), a ring

~

A semiprime rin$

~

~

~F~

onto

~I~ j

for some ring

~

that

~'

nonzero socle.

By the ring analogue of 1.7.5, we may suppose that

in a maximal essential

extension

the proof of uniqueness

~

of the socle ~,

~"

6'

satisfies

in 7.3, it follows

X

of

~ = X

of

~

such that

£

and

BA

~'

Hence

is embedded ~ = ~I~'

~J~.

Similarly as in

(x~"~ R, r E ~).

~', we have that

which proves that

is said to be invertible

AB

3'

into

R' ~

T r~ = T rX, where

by 5.9 which by 1.7.5 yields is additive.

We now consider isomorphisms of Rees matrix rings. division ring

ac-

that

is a dense extension of

Consequently

Since

the conditions of 7.4 and ~ e n c e

(xg)(r~) = (xg)(rk)

be the semigroup socle of But

[4].

is a semiprime ring essential extension of its

to an isomorphism

(r~p)(xg) = (rk)(x¢),

~'.

has can be characterized m u l t i p l i c a t i v e l y

is the semigroup socle of

admits a unique extension

£

An I X I'-matrix

B

if there exists an I' X l-matrix

over a A

over

are the identity matrices of the c o r r e s p o n d i n g sizes.

The following p r o p o s i t i o n is due to Hotzel

[l], its proof appears here for the first

time. 11.7.6 PROPOSITION. matrix

rings.

X = (xij)

Let

over

h

~

Let

~ = ~(I,h,A;P)

be an i s o m o r p h i s m o__ff g

we write

finite A × A ~ - m a t r i x over &', an___ddsuppose that

X~ = (xij~).

h', B

P$ = AP~B.

i s an i s o m o r p h i s m o f

~

onto

and

~' =

onto

~(I',hJ,A';P ')

Next let

A

be an invertible row

~J,

Then the function

X

defined by

(X E ~) Conversely ever~ isomorphism of

can be so constructed. To prove the direct part, we let (X*Y)X

= B(X*Y)$A

be Rees

g', and fo____rran____yymatrix

be an invertible column finite I'X I-matrix over

x:X ~ B(X$)A

PROOF.

X"

extension of its nonzero socle, has unique addition.

be an isomorphism of that

r~0 = r k.

and

which has no ideal which is s division

it follows

Rt

e

(snd is thus a

cording to 5.9,

T = T(~":R').

imply that

[i], see also Gluskin

a ring with the properties

Letting

~¢.

has unique addition if and only

isomorphism onto another ring is additive

rin$, a n d is an essential

~

into

by the uniqueness of both

The next corollsry is due to Rickart

II.7.5 COROLLARY.

where

~

follows immedistely from 7.3.

ring isomorphism).

Let

to an isomorphism of

~, then

and hence

Recall that by 1.5.22,

PROOF.

9

X,Y 6 •

and obtain

= B(XPY)$A

= B(X~)(P$)(Y$)A = B(X$)(AP'B)(Y$)A

(XX)P'(Y X) :

(X~) * (Yx) ,

~

onto

~

109 Since B

X

is obviously

easily

implies

additive,

that

For the converse, division assume

ring,

that

k

maps

we let

the statement

~

it is a homomorphism. ~

k

onto

~'

be an isomorphism

in the proposition

is not a division

The imvertibility

of

A

and

and is one-to-one. of

~

onto

is trivially

ring and consider

~'.

If

satisfied.

the commutative

~

is a

We thus

diagram

X

au(a,v) . where

the vertical

2.8, and of

{(~,a)

(U,h,V)

the isomorphisms

is the isomorphism

onto

of describing exploiting

arrows denote

(U',h',V')

in the proof of "iv) = vi)" in

induced by the semilinear

provided by 1.5.13.

the isomorphism

the commutativity

, au, (k' ,v')

X

isomorphism

(~,a)

The idea of the proof consists

as in the statement

of the proposition

by

of the above diagram.

Let

z=iz~IxeA], z' = { z ' t be bases of

V, U, V'

I X ' E Ar}

I

and

w'={w~,li'E

U ~, r e s p e c t i v e l y .

(m,a)

linear isomorphism

w=iw iliE f], Let

I' } b

be t h e a d j o i n t

of

the

semi-

and l e t

A = (ceXX,)kEA,X,EA,,

B = (~i,i)i,Ei,,i~i

where =

~ z' k'6h '~kk' ~''

z a Then

A

is row finite,

size and are invertible

B

is column

since both

b-lw i =

finite and both a

and

b -I

(1)

~ w' i'£f' i'll'i" A

and

B

are one-to-one

have the required and onto.

Further,

we obtain Pki w = (Zk,Wi)tO = (zk,b(b-lwi))~

= ( ~

~,z'

X'EA'

r

= (z a,b-lwi )I

w'

i'' i'~l'

'

i'~i~i)

Wj I k'' i ') ~i'l

Z /

=

= which

~ X'EA'

~ ~k'( i'El I

E

in terms of matrices We also have

E

zi,s -I =

(2)

'Px

]~'6A' i'{l '(~xk

''i'~i'i

has the form ~ %E A °k 'Xz k

P~ = AP'B

as required.

so that by (1)

we obtain

110

z!

z' a -I ~EA

= ~-~ ( ° k ' x '~) lEA which

XEA

£~ , ~ k ~ ' z "

~

~' i

•

~

(~

~'EA'

,)z',

(o, , . ~ ) ~

)EA

~ ~

~

implies

where

(o%,X~)~k , = 61, ~, (t',~' E i'), (3) X£A i s the Kronecker d e l t a f u n c t i o n . I t follows from (2) and (3) that

6k~p/

¢

XEA ~'6A' i

XEA

I' !

~'EA'

i'61' ~EA

A

(4)

= i'EI' ~J P'' i i ~ i 'i" Finally

let

X = (xix) E ~{. X

~

PX

=

Following

( Y], p . . x .

).

iEl ~i ~ and hence we must express

(z{,a-l)ea

a

-i

ca

the above diagram,

A,D6 i

in matrix

~

e

we have

a -I ~

form.

(5)

ca Using

(i) and (4), we obtain

= (~ol,k(zkc))~ lEA = ( ~ y s t e m of

0

Assume that

consistin$ o f hyperplanes

and let

HEY

u ={f~ Then

U

is a t-subspace o f

v*l f

is continuous]

V*, T = Tu(V )

n U = { f E V* I fZ ~ A H. -i=l l PROOF. V = f m @ Iv]

First let

for any vector

6f -I= [ x E Since

H

H 6 ~.

Vlxf

and

for some

H i E ~].

By 1.14, we have

v ~ fm

Then for any

= ~] = { y + o v l y E

is open so is its translate

H = f~

H+qv

for some

f ~ V*

and thus

6 E g, we obtain

H, (~v)f = 6] = for any

o E h.

U (H+~v). But the union of open

118

sets is again open w h i c h implies that In view of 1.17, U

so that

-i

is a subspace of

implies the existence of f E U

6f

vf ~ 0

H t ~

ix open. V*.

such that

which proves

Let

v ~ H.

that

U

Hence

f

is continuous.

0 # v E V.

The hypothesis

By the above, H = f~-

is a t-subspace of

for some

V*.

-i For any H

H E ]~, we also have

is Tu(V)-open.

topology

Since

~

But the sets

"ru(V )

topology If

um

so t h a t

f E U, then

for some

f ~ U

which shows that

is a subbase of the neighborhood system of

T, it follows that

• -open.

H = fz = Of

~ ~ Tu(V ) .

Conversely,

if

u E U, then

0

u ± = 0u -I

form a subbasc of the n e i g h b o r h o o d system of

"ru(V ) ~ "r.

f~ = 0f -I

Consequently

for the

0

is

in the

T = "ru(V ) .

is an open neighborhood of

0

and hence must

contain a basic open set ~ H. with H i ~ ~. Conversely, let f E V* and assume i= 1 l n fi ~ U as above. that fz ~ N H i for some H i 6 ]~. Then H i = f+i for some i=l n Hence f~ ~ ~i f.~ = Ill,f2,... fn ]m But then -- i=l i ' . f E

[f]~" = f~C__ Ill,f2,...,fn ] ~ -

= [fl,f2,... ,fn] c_ U

in view of 1.15. 111.1.19 LEMMA.

Let

S

be a proper subspace of a vector space

and extend

A

to a basis

a basis of

S

a basis of

V/S.

Conversely,

if

A

B

o_~f V.

is a basis of

is s_ system of r.epresentatives of the cosets of is a basis of PROOF. then

v =

S

Then S, C

C = ib+S

V.

is basis of

makin~ u_£

Let

A

I b ¢ B\A} V/S

C, then

and

b_!e is D

B = A U D

V. Let

A, B

>-~ m.

of complete

x,y C V, there exists

there exists

such that

m > ml,m2,m 3

whenever

(pointwise

v E V, there exists

n >_ ml; m3

is equivalent

v E V

Hence by Kelley

is complete,

to some point

product

o,T E &

exists

converges

to some

that for every For any

By Kelley

converges

for every

whenever

&V.

V*

which

{ fn I n h D} vf

product

to show that

of

V*

it is complete.

for all

in

V

while

t E T]

is a uniform neighborhood in U*. Since f is one-to-one, we conclude that both -i f and f carry uniform neighborhoods onto uniform neighborhoods, proving that both are uniformly

continuous.

dense subspace

of

U*, where

completion

of

(V,Tu(V)).

completion

is unique

For

nat V = U*.

f

is a uniform isomorphism

the latter is complete.

Since both

up to a uniform

III.4.3 COROLLARY. if and only if

But then

(U,V)

(V,~u(V)) isomorphism a dual ~ ,

and

Thus

by Kelley V

(f,(U*,~))

(U*,~)

of

V

onto a

is a

are Hausdorff,

the

([I], p. 197).

is complete

in the U-topology

131

111.5

L I N E A R L Y C O M P A C T VECTOR SPACES

We consider here linearly and weakly topologized and linearly compact vector spaces and establish relationships III.5.1 DEFINITION. M

If

M

is a topological module if

of these notions with those already encountered.

is s module

~

(left or right) over a ring

is a topological ring, M

under addition and the action of

~

upon

M

is jointly continuous.

is a linearly topologized module if the topology of neighborhood system of

0

M

consisting of submodules.

~,

then

is a topological group Further, M

admits an open base for the A topological vector space is

weakly topologized if it has an open base for the neighborhood system of

0

con-

sisting of subspaces of finite codimension. We are interested here in Hausdorff

topological vector spaces over discrete

division rings. As a supplement

to and partly a consequence of 1.21, we have the following

result. III.5.2 PROPOSITION. (V,T)

The following conditions on a topological vector space

are equivalent. i)

ii) iii)

V

is weakly

T = Tu(V ) V

topologized.

f o r some t-subspace of

V*.

is linearly topolo$ized and every open subspace has finite codimension.

PROOF.

Items i) and ii) are equivalent by 1.21.

i) = iii).

Obviously a weak topology is linear;

the second part of iii) follows

immediately from 2.7. iii) = i).

Each subspace in an open base of the n e i g h b o r h o o d system of

hypothesis must have finite codimension, The next result further elucidates weakly topologized spaces

If

linear forms on

PROOF. V*.

By 1.17, U

Let

~

subspaces of

V

Since

0 ~ x ~ V, there exists containing function extend

x f

f

is linearly

•

D

linearly

by

V*.

is a subspace of

T

V*.

We show next that

is Hausdorff, we infer that

B t ~

such that

by letting: vf = i to all of

V.

then the set

Furthermore,

U

of

Tu(V ) ~ T

is a weak topology.

and whose intersection with on

topologized,

is a t-subspace o f

be an open base of the neighborhood system of

V.

0

is indeed weakly topologized.

the relationship between linearly and

(&,V,7)

and the equality holds if and only i f

of

V

(cf. 1.18).

III.5.3 PROPOSITION. all continuous

so

if

For any

x ~ B. B

Let

U

0

[i B = 0. D

is a t-subspace

consisting of Hence for any

be a basis of

forms a basis of

v ~ D\B, vf = 0 o ~ &, we obtain

if

B.

V

Define a

v E D [I B, and

132

of -I = iv E V I vf = o] : [y + z I Y E B, z ~

[D\B],

zf = o]

~ [D\B] I zf = O} =

= B+[z

(B + z) U z E [D\B] zf=o

which is a union of open sets and is thus open. and thus so

fu

that

U

is a t-subspace of

V*.

is continuous and 1.16 yields T

is a w e a k topology.

and 1.18, we have linear forms on

V.

T = 7uI(V) = Tu(V )

Since

Tu(V ) ~

Conversely,

T = Tu~(V )

where

U

let

U~

V/S

M,

then

PROOF.

for every

M/N

Let

# N.

If

[K~c~A

N

if

~:M - M / N

V

is weakly

0

it follows that

K s~ + N.

+N

topologized and

S

is

The next result

for linearly topologized modules.

N

in

M

III.5.5 DEFINITION. m+N

But then

0

in -I

and thus K~ ~ F~ -i P~ and hence K ~

If

N

M/N.

is closed,

m ~ K~+N.

0 Then

is a union of open sets

an open base for the neighborhood system of

$

and thus

be the canonical homomorphism.

~ an open submodule of

and since

so that

n e i g h b o r h o o d of

A family

1.21

is a closed submodule of a linearly topolo$ized

fi K ~ = 0 in M/N. ckA ~ Let P be an open neighborhood of

then the set

U = U~

is also weakly topologized.

K ~ = K

K m

m { N

N N = ~

we have

be an open base for the neighborhood system of

c~ E A, the set

Then

By 5.2,

i__sslinearly topologized in the quotient topolosY.

thus open w h i c h makes

M,

be a weak topology.

that the corresponding statement also holds

consisting of submodules and let

(m+K~)

T

xf = i u E U,

T = Tu(V ), then 5.2 implies

also has this property,

V, then

111.5.4 PROPOSITION.

m+N

If

with

for any

is the vector space of all continuous

Note that 2.12 can be rephrased thus:

module

T.

f E U

fu = u

aS required.

a closed subspsce of asserts

Consequently

With this notation,

m~ ~ K ~

M/N.

in

K~

Then

P~

-i

~ ~ A.

which proves M/N m

such that

which proves

for some

is a submodule and

is a linear variety,

Let

there exists

P 0

U (K + n ) and is hEN m E M be such that

that

is an open Since that

N ~ P~

[K TI]o~A__

-i

,

is

consisting of submodules. is an element of a module

or simply s subvariety,

of

M.

of sets has the finite intersection property if the intersection

of any finite number of members concept, due to Lefschetz

in

$

is nonempty.

We now come to a fundamental

[i]; it represents a w e a k e n i n g of the n o t i o n of com-

pactness. 111.5.6 DEFINITION.

A linearly topologized module

M

is linearly compact if

every family of closed subvsrieties w h i c h has the finite intersection property has nonempty intersection.

133 111.5.7 PROPOSITION. M,

then

M/N

PROOF.

By 5.4, M / N

closed subvarieties of ~ : M ~ M/N

If

N

intersection,

M

Let

iC }0~ A

be e family of

having the finite intersection property and let Then

tC~ ~-I~jo~A

is a family of closed

having the finite intersection property and thus has a nonempty

say

C.

~ #

Thus

c~

( fi C

=

-i)~ ~ fi c ~ -i u = fi c -- o~A ~ c~:A ~

o~A M/N

topology.

is linearly topologized. M/N

be the canonical homomorphism.

subvsrieties of

and hence

is a closed submodule of a linearly compact module

i__~slinearly compact in the quotient

is linearly compact.

III.5.8 PROPOSITION.

A discrete linearly compact vector space is finite di-

mensional. PROOF.

Let

be s basis of

V

V

be a discrete and infinite dimensional vector space.

and for every

x E B, let

S

be the subspace of

V

Let

B

generated

X

by

B\[x] .

For . any .

.Xl,X2,

,x n ~ B, we obtain

(x I + x 2 + ... + x i _ I) + x i + (xi+ 1 + ... + x n )

E x i+S

xi

n

so

i-l~(xi + S x ' )

# ~"

Since

V

is discrete,

the family

IX+Sx]x~ B

consists of

i

closed subvsrieties and has the finite intersection property. where

xi,x E B.

n m ~ ~ixi = x+j~__l~jZj i= I

Then

so that

Let

y =

~ ~ixi ~ x+S x i=l

n m ~ ~.x - ~ ~.zj i= I i i j=l j

x =

where

z

E B and x ~ z for 1 < j < m. By linear independence we must have x = x i J J -_ , for some i. It follows that y ~ x + S x for all x # x i for i < i < n and thus ~L ( X + S x )

= @.

Consequently

V

is not linearly compsct.

x6B A family if for each

~

of subsets of a uniform space

~ C ~

there exists

next result is due to Dieudonn~ III.5.9 THEOREM. PROOF.

F E ~

(X,~)

such that

is said to contain small sets

F c X[x]

for some

x E X.

The

[5].

Every linearly compact module is complete.

In view of Kelley

([i], p. 193), it suffices to show that every family

of closed sets w h i c h has the finite intersection property and contains small sets has nonempty intersection. compact module

M

neighborhood system of exists

F E ~

It follows that x E

Hence let

~

he s family of subsets of a linearly

satisfying these requirements and let

such that

0

consisting of submodules of F ~ K[x]

F ~ iY E M I x - y

for some e K]

~ M.

x ~ M~ where

which, we claim,

be an open base for the For every

K E E, there

~ = ~(y,Z) l y - z ~ K } .

is equivalent

to

N (f+K). For if F ~ [y E M I x - y % K}, then for any f ~ F, x - f 6 K so fE F x E f+K and thus x E A ( f + K ) . Conversely, if x E ~I ( f + K ) , then x ~ f + K f6F fEF

134 for every

f E

F, so

x - f 6 K

and

thus

f t ty ~ M I x - y L K]

for every

f ~ F.

Let

We have

just

Let

F+K

x+K

~

seen

~ Q

that

and

f~ F K ~ ~{, there exists

for every

x C

N

(f+K).

Then

for any

F ~ ~

such

f 6 F, we have

that x E

F+K

f+K

E ~. and thus

fEF F+K.

k' E K.

If

f C F,

For any

k 6 K, we

f+k so that

F+K~

=

If

F+K

$

has

n

closed, H 6 $

there such

x = h+k z ~ M

that

that

which

quently

Then

~ q,

that

K

F+K

is open and since = x+K

is a

then

such

that

(x+K)

For

Now

Since

N F = ~,

contradicting x ~

this

K

iet

g t H.

F

there

x 6 H+K,

implies

= h + k+ K = x+K

Therefore

logical rin~ if the additive

is a topological group and the multiplication is jointly continuous,

be denoted by

PROOF.

cb =

This gives an intrinsic characterization of the topology

tle(a ) = ib ~ ~u(V) I ca = cb}. ~ c (a) = B(v~

u E U

B(v l,Vk2 ,...,vXn)(O) = ~r(C).

that the right annihilators of elements of neighborhood system of

all

Let

d E ~c(a)

We also require that (~u(V),#u(V))

be Hauadorff.

is a topolo$ical ring.

a,b ~ ~u(V), c ~ ~u(V) and

T

to

and consider the basic open set

~c(ab).

e C flca(b), we obtain c(de) = (cd)e = (ca)e = (ca)b = c(ab)

which shows that

de ~ ~c(ab).

Hence

plication is jointly continuous.

~c(a)~ca(b) ~ ~c(ab)

~c(a) -~c(b) ~ ~c(a -b), proving that the additive group of group.

Suppose next that

a # b.

by 1.3.3 implies the existence of ca # cb; if

Then

and the topology

a -b

c E ~u(V)

d ~ ~c(a) n ~c(b), then

~c(a) ~ ~c(b) = ~

so that the multi-

A similar argument shows that ~u(V)

such that

c(a -b) # 0.

ca = cd = cb, a contradiction. ~u(V)

is a topological

is a nonzero element of

is Hausdorff.

~u(V) Hence Thus

which

138 III.6. 4 LEMMA.

In

~u(V),

we have

i)

~Z(~T(V))

= ~ ( T ~)

if

T

is a subspace o f

U,

ii)

~r(~u(S))

= ~s~(V)

if

S

i_.~s~ subspace o f

V.

PROOF. suppose

We will prove i); the proof of ii) is left as an exercise.

that

U

is a t-subspace

sequence of equivalent

(bU* ~ T

~

b E ~u(V),

v

6 V

b*U ~ T

Item i) follows

a~T(V ) = 0 ~

implies

Only the last equivalence and

V*.

from the following

statements

a E ~%(~T(V))

u.i E T

of

We may

ab = 0) ~

needs a proof.

be such that and

ab = 0

b E ~T(V)

Va ~ T m. bU* ~ T

Suppose =

(v~,u i)

b*u i = u i.

for all

V # 0

Hence for any

implies

and let v ~ V

ab = 0.

Let

b = [i,y-i,x]. using

Then

the hypothesis,

we obtain (va,ui) = (va,b*ui) and since

= (v,(ab)*ui)

= (v,O) = 0

u. % T is arbitrary, we have (vast) = 0 for all t E T i Conversely, suppose that Va ~ T ~. Then for any b E £u(V)

Va ~ T ±.

b*U ~ T, we immediately since

= (v,a*b*ui)

b*u E T

and

obtain

Va c T m.

III.6.5 COROLLARY. we have in

for any Thus

For any

u E U, v E V, (v(ab),u)

v(ab) = 0

c E gu(V)

for all

and

v E V

so that for which

= (va,b*u)

= 0

and hence

ab = 0.

[Vl,V2,...,Vn]

a basis of

Vc,

£u(V),

~(vc)~(v) = ~r(~(vc)) = B(v l,v 2,...,vn)(O) = ~c(0) = ~r(C). Note

that in view of 2.7,

a base for the neighborhood

the family of all open subspaces

system of

0; an analogous

situation

of

V

constitutes

occurs

in

~u(V),

viz. III.6.6 COROLLARY. of

~u(V)

constitutes

topology

T

of

U

V.

~T(V)

IT

open subspace

of

UJ

system of

of right i d e a l 0

for the

#u(V).

PROOF.

of

The family

an open base for the neighborhood

Interchanging

the roles of

is open if and only if For each such

from 1.2.3-2.5. the family

~r(C)

S

T = S~

there exists

Hence by 6.5, where

U

c

and

V

is 2.7, we infer that a subspace

for some finite dimensional c ~ ~u(V)

such that

the family in the present

varies over

Vc = S

corollary

subspace

S

as follows

coincides

with

~u(V), which by the above discussion

has

the required properties. III.6.7 EXERCISES. i)

Find necessary

and sufficient

conditions

in order that

~u(V)

be discrete.

139 ii) S

of

For a dual pair

~u(S),

and finite dimensional ~u(V)

subspaees

T

relativized

to

U

and

~T(V)

of

and

respectively?

iii)

Let

bilinear

(U,V)

satisfy all the conditions

form may be degenerate•

the bilinear iv) T~ =

(U,V)

V, what can be said about the topology

Let N

~r(v) V)

What are necessary

form in order that (U,V)

of a dual pair except

~u(V)

be a dual pair.

and sufficient

that the

conditions

on

be Hausdorff. Show that for any subspace

T

of

U, we have

Na .

Show that

(~u(V),$u(V))_

has no proper closed ideals.

For the origin of the results of this section as well as for further discussion, see Jacobson

[5],([6], Chapter

IX, Section 6),([7], Chapter

111.7 The topology in question

is the topology

henceforth will not be mentioned properties or

U

of one-sided

~.

there exists

Let

for some

#u(V)

relativized

to

~u(V)

and

We study here a few topological

The closure of s subset

c E ~u(V).

such that

u.1 6 U, we have

a 6 ~(S).

left ideal of

and

Consequently

(~v)a and thus

~u(V)

The next two propositions

Eyery

a E ~u(S)

b E ~(S)

= Veb ~ vb ~ S.

~u(V).

18).

A

of

~u(V)

• t are due to D1eudonne

[2],

[5]•

III.7.1 PROPOSITION. PROOF.

explicitly.

ideals of

will be denoted by

see also Jacobson

A TOPOLOGY FOR

IV, Section

cb = ca.

Vca ~ S

Hence

III.7.2 PROPOSITION.

~(S)

is closed.

hc(a ) Fi ~ ( S )

Hence

for all

(vh,ui) = ~ ~ 0 = ~vk[i,o-l,~]a

~u(V) Then

Vb ~ S

c E ~u(V).

# ~

and hence

so that Let

~v

Vca ~ V; then

so that

E S

c ~u(S).

For every right ideal

R

o__ff ~ = ~u(V), we have

= ~r,$~%,~(R). PROOF.

First note that

~r,~,$(~T(V))

Let

= ~T ~m(V)

= ~ (V) by 6.4 and 2.2. T independent vectors in V. Then

a 6 ~_(V) and Xl,X2,...,x n be linearly T s*U ~ ~ and letting S = [Xl,X2,...,Xn], for any

u 6 U, we have

Further, we can write

minimal.

a =

~ ' k=l [~k'Vk'kk ]

with

m

( a * u + S ~) A T # @.

It follows

that

m (k~=lUik~k(Vkk,U) +S±) •

By I 2.3,

there exist

t

/

Ul u2''"

.

u

!

N T # ~

' m £ U

such that

(u 6 U). (vAi,u!)j

=

6iJ

for

1 _< i,j _< m

140 since infer

V%l,VX2,...,VXm

are linearly

of

the existence

g k E S±

independent

such

that

by 1.2.5.

Computing

t k = U i k ' Y k + g k E T.

a Uk, we

Letting

m t k = u .Jk "rk

and

b ~ ~ l[Jk,Tk,Xk] " , we obtain

m ~ u. T. (v

b*u =

k=l

]k k

m ~ t (v

,u) = )'k

k= 1 k

,u) E T )~k

and m

Xpb = k~__l(Xp,Ujk)TkV x k = k~= l (Xp ' t k ) v/"k m ~ (x ,u.

=

k=l for

1 < p < n

k ~'k

B(Xl,X2,.-.,x n)

(a) fi ~T(V)

(1)

a E ~T(V).

Conversely, vectors

= Xpa

)~.v

~k

so that

b E and thus

P

in

V

Xkb = xka

let and

for

a E ~T(V).

Let

u ~ U, Xl,X2,...,x n

S = [Xl,X2,...,Xn].

1 ~ k < n

and

There

b*u ~ T.

Let

exists

be linearly

b

t = b'u;

satisfying then

independent

(i),

so

t = a ' u + (b ~ a)*u,

where (Xk,(b - a)*u) = (Xk(b - a),u) = (0,u) = 0. Thus

t C

( a * u + S m) N T

which

proves

that

a*u E ~

and hence

a*U ~ T,

that is

a E g_(V). T III.7.3

COROLLARY.

For any subspace

gT(V)

= ~_(V) T

It is clear that the set is always

a right

generated III.7.4

by

ideal,

= ~ ~ T

of

U, we have

(V) = ~ , ~ , ~ ( a T ( V ) ) .

~r,~(A),

and that

T

where

~r,~(A)

A

is a nonempty

= ~r,~(L),

where

L

subset

of a ring

A.

COROLLARY

c(~.

2.5).

Th____eefollowing

conditions

on a subspace

T

of

are equivalent. i)

T

is closed.

ii)

~T(V)

is closed

iii)

~T(V)

is ~ right

PROOF.

This corollary subspaces

of

corollary

is valid

111.7.5

annihilator.

Exercise.

U

~,

is the left ideal of

implies

onto closed

that the lattice right

ideals

for open subspaces,

COROLLARY.

isomorphism

of

Su(V).

see 7.10,

Every principal

risht

X

in 1.3.10 maps

A slight variation

exercise

ideal of

iv). Su(V)

closed

of this

is closed.

U

141 PROOF.

Exercise.

An important kind of right ideal is provided by the following. III.7.6 DEFINITION.

Let

R

be a right ideal of a ring

is a modular

left identity of

~

relative to

such a case

R

R

if

~.

r - ar E R

An element for all

a E

r ~ ~; in

is a m o d u l a r risht ideal of

The next three results are new. 111.7.7 PROPOSITION. if

T

A right ideal

~T(V)

o_~f ~u(V)

contains an open subspace. PROOF.

r E ~u(V). u = uto.

Necessity. Then

s =

For every

By hypothesis

r - ar E ~T(V)

n

There exist

and

a E ~u(V)

and all

u E

and

(Va) ~

n

Vl,V2,...,v n E V

vk

Let

(k~=l(V,Uik)NkV~k,Ut) = k=l ~ (v,ulk)~k(VXk,Ut) . such that

(v.,u. ) = ~jk J ~k

1.2.3, and hence the last equation implies ~

for some

~ [ik,Nk,~k] , w h e r e n is minimal. k=l v E V, we have (va,u) = 0 and thus

0 = (va,ut)=

Let

is modular if and only

be such that

that

N ( v k , u t) = o.

u = ut~ = ut~(v

for

i ~ j,k < n ---

(Vkk,Ut) = 0

for

by

k = 1,2,...,n.

Then

,ut) n

= [t,~,k]u t - ( ~ [ik,~k,kk])[t,~,~.]u t E T k =i = by hypothesis.

Consequently

Sufficiency.

Since the subspaces

the n e i g h b o r h o o d system of

dimensional subspace Ve = S

implies

S

For any

V. Thus

dim Va < oo

S ~, where

0, the hypothesis

of

(e.g., use 1.2.3). u E T.

(Va) ¢ c_ T, where

implies that

(Ve) ¢ C T, that is and

(Va) m

is o p e n

dim S < ~, form an open base for S mC

There exists an idempotent

v E V, u E U

and thus

T

for some finite

e ~ ~u(V)

(ve,u) = 0

such that

for all

v E V

r C Su(V), we obtain

(ve,(r - er)*u) = (ve,r*u - e*r*u) = (ve,r*u) - (ve,e*r*u) = (ve,r*u) - (ve,r*u) = 0 so that

(r - er)*U c_ T

III.7.8 COROLLARY.

and therefore ~u(V)

r - er E ~T(V).

has an idempotent m o d u l a r

left identity relative

to every m o d u l a r right ideal. PROOF.

This follows from the last part of the proof of 7.7.

111.7.9 T H E O R E M (cf. 2.9).

The followin$ conditions on a subspace

are equivalent. i)

T

is a closed hyperplane.

T

of

U

142 ii)

~T(V)

is a closed maximal risht ideal.

iii)

~T(V)

is a modular maximal risht ideal.

PROOF.

i) ~

i) = iii).

This follows from 7.4 and 1.3.4.

This follows from 7.7 in view of 2.9. n Let e = ~ [ik,~k,X k] E ~u(V) be such that k=l

iii) ~ i). r E ~u(V).

all

ii).

Let

u i E U\T

plane, it follows that

and

r = [j,6,v].

u. = t + u . T J i

for some

Since

t E T

T

and

r-er is

E ~(V)

evidently

T E A.

for

a hyper-

Hence

(r - er)*u = ([j,6,~] - k=l~ [ik,~k(Vxk,Uj)~,v])*u n - ~ u. ?k(V. , u . ) ) ~ ( v ,u) (uj k=l ik ~k J n n = I t - k=l~U.lk~,(vmA k ' t ) - ['~ luU iik ~ )k ( kV X k= for all

t E T, • E A, u E U. n

-ui]T}6(v~'u)

E T

Consequently n

u i Vk(V I ,t) + [ ~ u i Vk(V k ,ui) - ui]T E T k=l k "k k=l k k for all

t E T, T E A.

For

t = 0

and

T ~ 0, we obtain

n

(1)

k~=lUik~k(Vkk,Ui ) - u i E T and for

T = 0, n

k=l

(t E T).

u i Vk(V k ,t) E T k k

By 1.14, we have Hence

(1) yields

T = gZ for some g 6 U* relative n g ( ~ u. ~ , ( v ,u.) - ui) = 0 whence k= I i k K Ak i n gu i =

and b y ( 2 ) ,

(2) to the dual pair

(U,U*).

(3)

~ (gu i )~k(Vk ,ui), k=l k k

we o b t a i n n

(4)

(gu i )~k(Vk ,t) = O. Now s u p p o s e and

(4)

that

k=l

k

(Vxk,U)

= 0

k for

k = 1,2,...,n.

Then

u

gives gu = g(t +uiT ) = gt + (gui)T

= k=if(guik)Vk(Vkk,Ui,) = k=l~(guik)Vk(Vxk,U)

= (gui)T n

= k~=l(gUik)Vk(Vkk,U - t)

= O.

=

which by (3)

t+u.T i

143

It follows

that for

S = [fvxl'fvx2'''"fVkn]

g E by 2.3.

Consequently

T = g~ = v z

so that

S ± ~ gZ

[g] = g± ~ ~ S ~ ~ = S

g E nat V

and hence

, we have

T

and thus

is open since

g = fv

for some

v E V.

v ~ = fvlO, and thus

T

But then

is also closed.

III.7.10 EXERCISES. i)

Show that a maximal

ii) if

V

right ideal of

Prove that every right ideal of

~u(V)

~(V)

is either

closed or dense.

is a right annihilator

if and only

is finite dimensional.

iii)

Show that for any right ideal

R

of

Su(V),

we have

= ~a E ~u(V) I n N r c_ Naj. rER iv) of

For a dual pair U

v)

(U,V),

prove that

the following

conditions

on a subspace

are equivalent. a)

T

is open,

b)

~T(V)

is open.

c)

~T(V)

is the right annihilator

Show that in

~(V),

provided with

of some element

the finite

of

topology,

~u(V). every left ideal is

a left annihilator. vi)

In any ring

~

(not necessarily

(i - s ) ~

~r - ar

~}

Ir E

a E ~. Show that for e = e , we have (i - e)~ = ~r(e). 2 where 6 ~u(V), we have ( i - e)~u(V ) = $ ( l _ e ) , u ( V )

Also show thst

e = e

(i - e ) * U = ~ u - e*u l u E U} relative vii)

to

~T(V)

For any

viii)

Let

a E ~u(V)

The principal

we

let

Mr(S ) T

left identity

=

~(Va)X

for

~u(V)

(v).

be a subspsce

~T(V) = ~r(S)

for some

of

U.

Show that

a E Su(V).

for this section are Behrens [5],([6], Chapter

([i], Chapter

IX, Section 8),([7], Chapter

II, Section I, Section 3

18). III.8

~,

is a modular

(Ve) ± ~ T.

show that

references

[2], Jacobson

IV, Section

For a set

e

be a dual pair and

if and only if

7), Dieudonn~ and Chapter

and that

if and only if

(U,V)

codim T < ~

E

write

2

for any for

=

with identity),

of functions

ANOTHER TOPOLOGY on a set

X

FOR

£u(V)

into a uniform space

(Y,~),

for every

144

W(~) : [(f,g) E $ x $ Then the family uniformity

I (f(x),g(x)) E ~

[W(~) I ~ E ~}

of uniform

is a base

convergence;

for all

x E X].

for a uniformity

the corresponding

on

topology

$

called the

is the topology of

uniform convergence. We now apply this construction to the set i.

£u(V)

of functions

An open base

family

[T~ I T

2.

v E A

The uniformity

V 3.

for which According

convergence where

there exists

associated with we proceed

system of

U, dim T < ~}

~

on

V

0

in

Tu(V)

and

as follows.

V

is given by the

so that a set

A c V

a finite dimensional

induced by

for

on

Tu(V )

is a ~u(V)-neighborhood

~(A) = {(x,y) Ix - y ~ A].

on

Specifically

is open if

subspace

T

of

U

v + T z c A.

[~(A) I A where

V.

for the neighborhood

is a subspace of

and only if for every such thst

to the uniformity

on

Consequently

there exists

£u(V)

of ~

a neighborhood

to the above construction, induced by

~

has for a base the family 0}

consists A

of

0

of all binary relations such that

the uniformity

has as a base

~

~(A) ~

~.

of uniform

the family

[W(~) I ~ E ~ ,

~ = ~(A),

~('L((A)) = [ ( a , b ) E £u(V)) i,

Hence

~i

the socle of

is ~

equals

0. For further discussion of semisimple [2], Wiegandt

[2] and Eckstein

An obvious generalization compactness.

of linear compactness

Since every semisimple

socle (see Wiegandt

linearly compact rings, we refer to Leptin

[i]. is the notion of local linear

locally linearly compact ring has a nonzero

[4]), the study of their structure

is not the case in the theory of locally compact rings Linearly

compact nonsemisimple

[3] and Arnautov = ~(~)).

[i], and particular

[i], Fuchs

to Andrunakievi@,

Arnautov

(cf. Skornjakov

rings were investigated

by Leptin

[1],[2]). [2], Wiegandt

attention was paid to radical rings

Further results on linear compactness

Fischer and Gross

can be easily handled, which

[i], and Warner

were obtained by Wolfson

[i].

and Ursu [i], Warner

(where

For recent developments

[2], widiger

[i], we refer

[i] and Wiegandt

[5].

167

CURRENT ACTIVITY From the material

of these Lectures,

we may easily extract the following

categories: i.

Objects:

pairs of dual vector spaces,

Morphisms: 2.

semilinear

Objects: weakly Morphisms:

3.

Objects:

complete

lattices

(U-closed

Objects:

rings with a nonzero socle,

ring isomorphisms;

Morphisms: 5.

adjoint;

topologized vector spaces,

maximal primitive

Objects:

with a surjective

iseomorphisms;

Morphisms: 4.

isomorphisms

satisfying

subspaces of

the double covering condition

V),

lattice isomorphisms;

multiplicative

semigroups

of objects

in 3. above

(characterized

abstractly), Morphisms:

semigroup

isomorphisms.

Further categories may be obtained by taking certain subobjects Under some mild restrictions categories

are either equivalent

relationship

or isomorphic.

semigroups,

by the author entitled rings and lattices".

toward a better understanding

of homomorphisms

transformations, pierce

[i].

see Fajans

Multiplicative

[4], Satyanarayana peinado Eckstein

[I]. [2].

widiger

These Lectures

thus represent a step

among these different branches

of linear transformations

and antiisomorphisms [1],[2],

includes mainly in-

of various

semigroups

Jodeit and Lam [i], Mihalev

semigroups

Ill, a comprehensive

of linear

and Satalova

[i],

of certain rings have been studied by Petrich survey of this subject can be found in

Semigroup methods have been successfully

used in ring theory by

Further classes of rings with unique addition have been recently

found by Martindsle investigated

of vector and projective

geometry.

The recent work on semigroups vestigations

study of the

forms the subject of an un-

"Categories

of the relationship

of modern algebra an~ projective

in 3., 4., and 5.

most of the resulting

A comprehensive

among these and many other categories

published manuscript spaces,

on objects of these categories,

[i] and Stephenson

by Andrunaklevlc,

[i], Wiegandt

[5].

Arnautov

]i].

Linear compactness

and Ursu [i], Arnautov

for rings has been [i], Warner

[1],[2],

168

BIBLIOGRAPHY Andrunakievi[,

V.A., Arnautov,

V.l. and Ursu, M.I.

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V.I.

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3(1969),3-17

Baer, R. [i] Linear algebra Behrens,

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Academic Press, New York,

1952.

E.A.

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N.

[i] El6ments clifford,

1972.

de Math6matiques,

A.H. and Preston,

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IV, Hermann,

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1955.

G.B.

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J.

[i] La dualit6 dans les espaces vectoriels 59(1942),107-139.

topologiques,

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Ann. Ecole Norm. Sup. Bull. Soc. Math.

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N.J.

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Univ. Toronto Press,

1963.

F.

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18

169

Everett,

C.J.

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64(1942),363-370.

Faith, C. [I] Rings with minimum condition on principal 327-330. Faith, C. and Utumi,

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10(1959),

Y.

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theorem, Acta Math. Acad. Sci. Hung.

14(1963),

Fajans, V.G. [i] Isomorphisms of semigroups of invertible linear transformations leaving a conus invariant, Izv. Vyss. Ucebn. Zaved. Mat. 12(91)(1969),93-98 (in Russian). [2] Isomorphisms of semigroups of affine transformations, sibir. Mat. Z. 11(1970), 193-198 (in Russian); English transl, by Consultants Bureau 11(1970),154-158. Fischer,

H.R. and Gross, H.

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forms and linear topologies,

I, Math.

Ann. 157(1964),296-325.

Fuchs, L. [i] Note on linearly compact abelian groups, 433-440.

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Fuchs, L. and Szele, T. [i] Contribution to the theory of semisimple 3(1952),233-239. Gewirtzman,

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L.

[i] Anti-isomorphisms of the endomorphism Math. Ann. 159(1965),278-284.

rings of a class of free modules,

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Math.

L.M.

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Usp. Mat.

Izv. Akad. Nauk SSSR 22(1958),439-448

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Mat. Sbornik 47(1959),111-130

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(in

170

Goldman,

O.

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P.A. and Petrich, M.

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chain condition,

of semigroups,

Pacific J. Math.

26(1968),493-508.

E.A.

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Doklady Akad. Nauk SSSR 96(1954)~ U~. Zap. Ivanov.

Ped. Inst. 5(1954),

Hail, M. [1] Th____eetheory of groups,

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1959.

Hotzel, E. [1] On simple rings with minimal one-sided Amer. Math. Soc. 17(1970),238. Jacobson,

ideals

(unpublished).

Abstract:

N.

[1] Normal

semi-linear

transformations,

Amer. J. Math.

61(1939),45-58.

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finiteness

for arbitrary

assumptions,

1943.

Trans. Amer.

rings, Amer. J. Math.

67(1945),

rings, Ann. Math. 48(1947),8-21.

in abstract alsebra,

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Notices

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p.

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de certains

anneaux,

C. R. Acad. Sci.,

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Jans~ p. [i] Rin2m§ @ n d

homology,

Holt, Rinehart

and Winston,

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1964.

Jodeit, M. and Lam, T.Y. [i] Multiplicative Johnson,

to the theory of centralizers,

20(1969),10-16.

Proc. London Math. Soc. 14

R.E.

[i] Rings with unique addition, Johnson,

Archiv Math.

B.E.

[l] An introduction (1964),299-320. Johnson,

maps of matrix semigroups,

R.E. and Kiokemeister,

Proc. Amer. Math.

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F.

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Trans.

171

Kaplansky, I. [i] Topological rings, Amer. J. Math. 69(1947),153-183. [2] Locally compact rings, ~ner. J. Math., Part I, 70(1948),477-459; 73(1951),20-24; Part III, 74(1952),929-935.

Part II,

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1966.

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MacLane, S. [i] Extensions and obstructions

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Martindale, W.S. [i] When are multiplicative mappings additive? 695-698.

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172

Morita, K. [I] Category-isomorphisms and endomorphism rings of modules, Trans. Amer. Math. Soc. 103(1962),451-469. Muller, U. and Petrich, M. [i] Erweiterungen eines Ringes durch eine direkte Summe zyklischer Ringe, J. Reine Angew. Math. 248(1971),47-74. [2] Translationshulle and wesentlichen Erweiterungen eines zyklishen Ringes, J. Reine Angew. Math 249(1971),34-52. Neumann, J. von [i] On regular rings, Proc. Nat. Acad. Sci. 22(1936),707-713. Ornstein, D. [I] Dual vector spaces, Ann. Math. 69(1959),520-534. Peinado, R.E. [i] On semigroups admitting ring structure, Semigroup Forum i(1970),189-208. Petrich, M. [i] The translational hull of a completely 0-simple semigroup, Glasgow Math. J. 9(1968),i-11. [2] The semigroup of endomorphisms of a linear manifold, Duke Math. J. 36(1969), 145-152. 13] The translational hull in semigroups and rings, Semigroup Forum 1(1970), 283-360. [4] Dense extensions of completely O-simple semigroups, J° Reine Angew. Math., part I, 258(1973),i03-125; Part II, 259(1973),109-131. [5] Introduction to semigroups, Merrill, Columbus,

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174

Wiegandt,

R.

[i] Uber linear

kompakte

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regulate Ringe,

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nilpotente

Poi. Sci.

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28(1967),255-260.

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left

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D.

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[2] Linearly

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compact modules

[3] Every linear transformation Soc. 5(1954),627-630.

is a sum of nonsingular

75(1953),79-90. ones, Proc. Amer. Math.

175

LIST OF SYMBOLS

(&,v)

1

v, (V,A), &+, V * = (V*,A)

2

g(v) = g(A,V), ~(V) = £(A,V), g(V), ~'(V),

dimV,

A•B,

rank a, N a

3

(U,V) = (U,a,V) = (U,/X,V;~), fu' nat U

4

F(~,v), £u(V), Fu(b,V), gU(~,v), ~n,U(b,V), a*

6

A\B, bX' DO{, S-, ~BA-, [i,'f,X] C(A),

7 ii

iB(A )

16

~2, ~ ( N ) ,

~r (g)

19

~Rn

22

Es , } ( U , V )

23

Ge S / I , J(a),

25

I ( a ) , ¢+, g (A,V)

27

(v) Q ~°(I,G,M;P),

28 Ln(A),

Qn,u(A, V)

29

Xv,(V), (®,a), ( a , v ) ~ (Z~',v')

34

(b ,w)

35

(u,b,v) ~ ( u ' , ~ ' , v ' ) , g~,u(b, v)

~(w,a)

38 43

~g, m v

44

C(S) ,£+

45 46

~, A, M, I

47

g, G(S), ¢9(S),

48

HK

60 + V

49 54 k

~(S), %a' IDa' rra' [[(S), T = T(K:S), ~k , p 2

k

55

S

56

~(s)

6O 62

AB, aB, Ab A n , G£,~(B), AB, A n, aB Ai iE I

65 ~r,~(B)

66 67 68

176 70

AB

74 75

k op, £, ~,

76

~(I,A,A;P) g

80 81 91

+G A i iE I

92

~es ~A

¢~

(a)

95

~ ,~ c~AC~ c~

96 97

rl s c~ a6A

102 (a) r x , X A, (X,~) c~A ~

112 •

,

S(~;y), B((~l,...,~n,X[ . . . , X n ) ¢~AIIX , B(vl, . .,Vn;~l, . . .

' Tu(V)

= ~U(A,V)

113

.,~n ), C(vi,.. .,v n;6l,...,~n )'

SJ-, T z, [A], [Wl,W2,..-,Wn],

v't, uZ

114 115

(b,v ,'0

119

codim S

127

a*

X[A], X[x], (xj~) (S,~), [S n In e D}, (f,(X*,%~*))

128 129 130 136

~U (V)' B(xl,...,Xn;Yl,...,yn)' B(xl,...,Xn)(a)

'O.c(a), 7

137

(~, "r)

W(~), "~(A), ~ ,

139 'r

144 145

p(ul, . . .,Un) (a), ~c(a)