Ring Constructions & Applications (Series in Algebra, Volume 9)

ALGEBRA

RING CONSTRUCTIONS AND APPLICATIONS

Andrei V. Kelarev

World Scientific

RING CONSTRUCTIONS AND APPLICATIONS

SERIES IN ALGEBRA Editors: J. M. Howie, D. J. Robinson, W. D. Munn Vol. 1: Infinite Groups and Group Rings ed. J. M. Corson et al. Vol. 2: Sylow Theory, Formations and Fitting Classes in Locally Finite Groups M. Dixon Vol. 3: Finite Semigroups and Universal Algebra J. Almeida Vol. 4: Generalizations of Steinberg Groups T. A. Fournelle and K. W. Weston Vol. 5: Semirings — Algebraic Theory and Applications in Computer Science U. Hebisch and H. J. Weinert Vol. 6: Semigroups of Matrices J. Okninski Vol. 7: Partially Ordered Groups A. M. W. Glass Vol. 9: Ring Constructions and Applications A. V. Kelarev

Forthcoming Vol. 8: Groups with Prescribed Quotient Groups and Associated Module Theory L Kurdochenko, J. Otal and I. Subbotin

SERIES

IN

ALGEBRA

VOLUME 9

R|

NG CONSTRUCTIONS AND APPLICATIONS

Andrei V. Kelarev Department of Mathematics University of Tasmania Australia

V f e World Scientific « •

New Jersey • London • Singapore • Hong Kong Sir

Published by World Scientific Publishing Co. Pte. Ltd. P O Box 128, Farrer Road, Singapore 912805 USA office: Suite IB, 1060 Main Street, River Edge, NJ 07661 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

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RING CONSTRUCTIONS AND APPLICATIONS Series in Algebra - Vol. 9 Copyright © 2002 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in anyform or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

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ISBN 981-02-4745-1

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to my children, Lena and Nadia

Preface

A number of diverse ring constructions have been applied in various areas of modern science, from coding theory, to cryptography, to logic, to number theory, to quantum physics and symbolic computation. The investigation of properties of ring constructions is a very large and rapidly changing research area. The following two criteria have been used in selecting material for this book. First, the author has tried to choose topics, preliminaries and proofs in order to prepare new researchers for work on directions where active investigation has been carried out recently and there are enough challenging open problems. Second, an attempt has been made to supply advanced specialists with complete concise information and references related to these questions and directions. This book is devoted to various ring constructions, their properties, and examples of applications. It would be really valuable to find a single viewpoint which makes it possible to achieve good understanding of many notions in a unified framework. The emphasis is on the fairly new concept of a groupoid-graded ring embracing and unifying a variety of constructions. Familiarity with this efficient tool makes it easy to understand properties of and similarities among other more special ring constructions, which at first glance may appear quite different. Groupoid-graded rings include as special cases many other ring constructions: polynomial and skew polynomial rings, Ore extensions, direct and semidirect products, matrix and structural matrix rings, Rees matrix rings, Morita contexts and generalized matrix rings, group and semigroup rings, skew group and semigroup rings, twisted group and semigroup rings, monomial rings, smash products, crossed products, group-graded rings, vii

Vlll

Preface

monoid-graded rings, path algebras, edge algebras, incidence rings, etc. Therefore groupoid-graded rings can be applied to the study of other less general constructions. This not only gives new results for several constructions simultaneously, but also serves the unification of known theorems. Groupoid-graded rings have been actively investigated for years, and many interesting results have been obtained in the literature. Several surveys on this direction have already been published, but there was no monograph collecting various results in a convenient form with concise preliminaries included for the reader. Therefore it is also desirable to fill this gap and summarize numerous contributions made during recent years. This is a new and rapidly developing area of research. Many new open problems have been recorded in various publications, which also provides an opportunity for postgraduate students to take part in the ongoing work. A book on this topic can serve as an excellent introduction for postgraduate students to many ring constructions as well as to most essential basic concepts of group, semigroup and ring theories used in proofs. Our exposition focuses on introducing sufficient background knowledge and preparing new researchers for work on several topics with abundant open problems and opportunities for new contributions. On the other hand, many more specialized, advanced and difficult technical results have been included without proofs but with references. In this way the text at the same time addresses the needs of advanced experts too. This approach has helped to provide valuable content and references for new and established researchers using ring constructions in their work. This book has grown from a series of publications by the author, which have been discussed with co-authors, colleagues, after the talks at various conferences and seminars, as well as with editors organizing anonymous refereeing of papers. The author is grateful to all mathematicians who have contributed to his research, and would like to use this opportunity to express sincere appreciation to Professors K.I. Beidar, A.D. Bell, S. Dascalescu, D. Easdown, K.R. Fuller, B.J. Gardner, S. Goberstein, T.E. Hall, K.J. Horadam, J.M. Howie, E. Jespers, J. Justin, R. Lidl, J. Meakin, A.V. Mikhalev, W.D. Munn, J. Okninski, F. Otto, D.S. Passman, F. Pastijn, G. Pirillo, C.E. Praeger, P. Schultz, L.N. Shevrin, I. Shparlinski, P. Shumyatsky, O.V. Sokratova, T. Stokes, M.L. Teply, P.G. Trotter, M.V. Volkov, A.P.J, van der Walt, R. Wiegandt, L. van Wyk, and to the Editor of World Scientific E.H. Chionh.

Contents

Preface

vii

Chapter 1 Preliminaries 1.1 Groupoids 1.2 Groups 1.3 Semigroups 1.4 Rings

1 1 2 4 11

Chapter 2 Graded Rings 2.1 Groupoid-Graded Rings 2.2 Semigroup-Graded Rings 2.3 Group-Graded Rings 2.4 Superalgebras

17 17 20 21 23

Chapter 3 Examples of Ring Constructions 3.1 Direct, Subdirect and Semidirect Products 3.2 Group and Semigroup Rings, Monomial Rings 3.3 Crossed Products 3.4 Polynomial and Skew Polynomial Rings 3.5 Skew Group and Semigroup Rings 3.6 Twisted Group and Semigroup Rings 3.7 Power and Skew Power Series Rings 3.8 Edge and Path Algebras 3.9 Matrix Rings and Generalized Matrix Rings 3.10 Triangular Matrix Representations

25 25 26 27 27 28 29 30 31 31 33

ix

x

3.11 3.12 3.13 3.14 3.15

Contents

Morita Contexts Rees Matrix Rings Smash Products Structural Matrix Rings Incidence Algebras

33 33 34 34 35

Chapter 4 The Jacobson Radical 4.1 The Jacobson Radical of Groupoid-Graded Rings 4.2 Descriptions of the Jacobson Radical 4.3 Semisimple Semigroup-Graded Rings 4.4 Homogeneous Radicals 4.5 Radicals and Homogeneous Components 4.6 Nilness and Nilpotency

37 37 42 45 51 64 67

Chapter 5

73

Groups of Units

Chapter 6 Finiteness Conditions 6.1 Groupoid-Graded Rings 6.2 Structural Approach of Jespers and Okniriski 6.3 Finiteness Conditions and Homogeneous Components 6.4 Classical Krull Dimension and Gabriel Dimension

77 77 82 89 102

Chapter 7

107

Pi-Rings and Varieties

Chapter 8 Gradings of Matrix Rings 8.1 Full and Upper Triangular Matrix Rings 8.2 Gradings by Two-Element Semigroups 8.3 Structural Matrix Superalgebras

111 Ill 120 129

Chapter 9 Examples of Applications 9.1 Codes as Ideals in Group Rings 9.2 Codes as Ideals in Matrix Rings 9.3 Color Lie Superalgebras 9.4 Combinatorial Applications 9.5 Applications in Logic

133 133 140 146 147 148

Chapter 10

149

Appendix A

Open Problems Glossary of Notation

153

Contents

xl

Bibliography

157

Index

201

Chapter 1

Preliminaries

In this chapter we give concise background information most frequently used in this direction of research for the convenience of the reader. Complete and detailed explanations of the algebraic concepts summarized below can be found, for example, in the following books: [Beachy (1999)], [Cohen and Cuypers (1999)], [Cohn (2000)], [Fan et al. (2000)], [Grillet (1999)], [Hazewinkel (1996)], [Howie (1976)], [Howie (1995)], [Hungerford (1980)], [Kargapolov and Merzljakov (1979)], [Lallement (1979)], [Lam (1999)], [Lambek (1976)], [Lidl and Niederreiter (1994)], [Lidl and Pilz (1998)], [Passman (1991)], [Robinson (1982)], [Rowen (1991)], and [Scherk (2000)]. Several more specialized and advanced monographs have also been included in the bibliography. 1.1

Groupoids

A set with a binary operation is called a groupoid. An associative groupoid is called a semigroup. A semigroup with identity is called a monoid. A groupoid G is called a quasigroup if the equation ab = c determines a unique element b G G for given a,c € G, and a unique element a € G for given b,c £ G. An element 1 is called an identity element or unity of the groupoid G if \g = gl = g, for every g £ G. A loop is a quasigroup with an identity element. Let G be a groupoid. An element 0 is called a zero of the groupoid G if Og = gO = 0, for every g e G. Denote by G 1 (and G°) the groupoid G with identity (respectively, zero) adjoined. If G has an identity element (or zero), then we assume that G1 = G (respectively, G° = G). Otherwise, l

2

Preliminaries

Gl = G U {1} (and G U {0}). If M is a subset of a groupoid G and x € G, then xM1 = xM U {a;} and M1x = Mx U {x}. UT Q G, then the subgroupoid generated by T in G is usually denoted by (T). It consists of all products of elements of T. Another notation is used in formal language theory. Namely, the subgroupoid (subgroupoid with identity) generated by T in G1 is denoted by T+ (respectively, T* ). The ideal (left ideal, right ideal) generated by T in G is the smallest subgroupoid J containing G such that GIUlG CI (respectively, GI C I, IG C / ) . If G is a semigroup, then the ideal (left ideal, right ideal) generated by T in G coincides with the set G^TG1 (respectively, GlT, TG1). An ideal generated by one element is called a principal ideal. For s € G, let s" denote the set of all x £ G such that s belongs to the ideal generated by x in G. The cardinality of the set G is called the order of the groupoid G and is denoted by \G\. The order of the element g is the order of the subgroupoid (g) it generates. An element g of a groupoid G is said to be periodic if the subgroupoid (g) is finite. A subset T of S is periodic if every element of T is periodic. An element e of G is called an idempotent if e = e 2 . The set of all idempotents of G is denoted by E(G).

1.2

Groups

This section contains a few most basic definitions and facts for the convenience of the reader. A semigroup G with identity 1 is called a group if every element g € G has an inverse g_1 such that gg~* = g~xg = 1. A commutative group is said to be abelian. Abelian groups are often considered in the additive notation where the operation of the group (G, +, 0) is denoted by + and the neutral element is 0. Let G be a group. A subgroup H of G is a subset closed with respect to the multiplication of the group and the operation of taking the inverse element. If T C G, then the subgroup generated by T is the smallest subgroup containing T. The set T is called a generating set of the subgroup. A subgroup is finitely generated if it has a finite generating set. The definitions of a finitely generated subsemigroup, or subring, or ideal of a ring are similar. If T is a nonempty subset of G and g e G, then the conjugate of T by g is the set T> = g-lTg

= {g'Hg

\teT}.

Groups

3

A subgroup H of H is said to be normal if the set HG =

{g-1hg\heH,g£G}

is equal to H. A right (left) coset of the subgroup H is a set of the form Hg (respectively, gH), where g £ G. The set of left cosets has the same cardinality as the set of right cosets, and this cardinality is called the index of H in G. If N is a normal subgroup, then the quotient group G/N is the set of all right cosets of N in G with the following operations: (Ngi)(Ng2) = iV;«,A)|/>eG,t€/}l

5 « = {(/ l ;i,A) SiX =

\h€G,XeA}, {(h;i,X)\heG}.

8

Preliminaries

If S = M°(G; I, A; Q), then we include zero in all of these sets, i.e., put S,x = {0} l) {(h;i,\)

\ h £ G,i e I},

Si, = {0} U {(/i; i, A) | h G G, A G A}, SiX =

{0}U{(h;i,\)\h€G}.

Theorem 1.8 Let G be a group, and let S = M(G;I,A;P) be a completely simple semigroup (or let S = M°(G; I, A; Q) be a completely 0-simple semigroup). Then, for all i,j G / , A, fi G A, h G G, and x = (h;i,X) G S, the following conditions hold: (i) (ii) (iii) (iv) (v) (vi) (vii) (viii)

the set S,x is a minimal (minimal nonzero) left ideal of S; the set Si, is a minimal (minimal nonzero) right ideal of S; Sx = S^x = S„x; xS = xSjt = Sit; x G Sx n xS = Six; the set Si\ is a left ideal of S^ and a right ideal of S*x; ifpxi = 0, then Sfx = 0; ifpxi 7^ 0, then Six is a maximal subgroup of S (maximal subgroup with zero adjoined in S); it is isomorphic to G (respectively, to G°); (ix) each maximal subgroup (subgroup with zero adjoined) in S coincides with Sj/j,, for some j G / , (J- G A; (x) M(G; I, A; P) is a right (left) group if and only if \I\ = 1 (respectively, |A| = 1); (xi) if S = M{G; I, A; P), then each S*x is a left group, and each Si, is a right group.

Theorem 1.9 If an epigroup S contains only a finite number of idempotents, then S° has a finite ideal chain 0 = SoCS1C...CSn

= S°

such that each factor Si+i/Sit where 0 < i < n — 1, is nil or is a completely 0-simple semigroup with a finite sandwich matrix. Theorem 1.10 If S is an epigroup without infinite descending idempotent chains, then S° has an ascending ideal chain 0 = 5 0 C 5i C . . . C UTST = S°

9

Semigroups

such that Sy = U^^S,, for every limit ordinal number 7 < r, and for each ordinal /x the factor 5' M+ i/S r /i is nil or is a completely O-simple semigroup. Theorem 1.11 alent:

For every epigroup S, the following conditions are equiv-

(i) S is Archimedean; (ii) S does not contain the two-element semilattice; (iii) has a completely simple ideal and the quotient semigroup modulo this ideal is nil. A linear semigroup is a semigroup of matrices with entries in a division ring. Denote by e%j or e,j the standard matrix unit with 1 in the (i,j) entry and all other entries equal to 0. The set {e„ I 1 rrrty = 0, for each t £ supp(r) and all x, y £ H(R). It is routine to verify that if the ring R is graded by a cancellative semigroup and J is an ideal of R, then all elements of minimal positive length in I are rigid. In particular, all homogeneous elements of J are rigid in this case. If G has a zero 0 and R is a G-graded ring with RQ = 0, then R is called a contracted G-graded ring. For any groupoid G, if I is an ideal of G and R is a G-graded ring, then the quotient semigroup G/I has a zero and R/Rj is a contracted G/J-graded ring. Similarly, if R = ®geoRg is a G-graded ring with a homogeneous ideal I, then R/I = © s 6 Gi? 9 /7 g is G-graded. A ring R = ®g^GRg is said to be strongly G-graded if RgRh = Rgh, f° r all g,h £ G. The grading of R is said to be non-degenerate if and only if the induced partial groupoid has an identity 1 and each of the equalities (T\R)I = 0 or (Rr)i

= 0 implies r = 0.

If R is a G-graded ring, then the largest homogeneous ideal contained in J(R) is denoted by J%T(R) or JQ{R) and is called the homogeneous part of the radical or the homogeneous radical. Let R = 0 s € G Rg be a G-graded ring. A left .R-module M = 0 s e G Mg is called a graded R-module if RgMh C Mgh, for all g,h £ G. Categories of modules over rings provide very relevant information on the structure of the rings (see [Anderson and Fuller (1992)]). This explains the interest in the investigations of properties of modules over graded rings. A graded left module is graded Noetherian if every ascending chain of

20

Graded Rings

graded submodules of the module terminates. Obviously, every left Noetherian module which is graded is also graded left Noetherian. The converse is not true in general (see [Dascalescu and Kelarev (1999)]). For a G-graded ring R, we denote by (G, R)-gi the category of G-graded i?-modules. If M is an object of this category, then the lattice of all subobjects of M in this category will be denoted by Lat(G ) fl)_ gr (M). Let K—dim(Gifl)_gr(M) denote the Krull dimension of this lattice (which is the same as the Krull dimension of M in the category (G, i?)-gr), whenever it exists. If M has Krull dimension as an i?-module, this dimension will be denoted by K—dimfl(M). The following properties of Krull dimension are well known. If N is a submodule of the .R-module M, then M has Krull dimension if and only if N and the quotient module M/N have Krull dimension, and in this case K-dim(M) = sup(K-dim(7V), K-dim(M/iV)). In particular, if M = 0 " = 1 Mi is a sum of a finite number of modules M» with Krull dimension, then M has Krull dimension and K—dim(M) = sup K—dimM;. i=\

2.2

Semigroup-Graded Rings

The Wedderburn's classical theorem tells us that every finite dimensional algebra over a perfect field is a semilattice-graded ring with two components: the radical and a semisimple subalgebra (see, for example, [Curtis and Reiner (1962), Theorem 72.19] or [Pierce (1982), §11.6]). The following interesting example is also due to Wedderburn. Every matrix ring Fn over a field F of characteristic zero contains idempotents £tj, where 1 < i,j < n, such that Fn = (BijFstj and SijSke € Feu for all 1 < i,j, k,£ s£S

s€S

\s£S

J \tes

seS

J

s,tes

If G is a group (monoid), then R[G] is called a group ring (respectively, monoid ring). If S is a semigroup with zero 6, then the contracted semigroup ring RQ[S] is the quotient ring of R[S] modulo the ideal R6. Thus Ro[S] consists of all finite sums of the form Y^7=iriSi> w n e r e rt £ R, 8 ^ s, S S, and all elements of R6 are identified with zero. If R is a field, then we use the terms semigroup algebra, contracted semigroup algebra, monoid algebra, contracted monoid algebra, and group algebra. Theorem 3.1 (Maschke's Theorem) Let F be a field, and let G be a finite group. Then F[G] is an Artinian ring which is semisimple if and only if char(F) does not divide \G\. Theorem 3.2 Let F be a field of characteristic p > 0, and let G be a finite p-group. Then the radical of F[G] is equal to the augmentation ideal n

LO(F{G\)

n

= {J2fi9i \fi£F,gi£G,YJfi i=l

= 0}i=\

Lemma 3.1 Let S be a semigroup such that the semigroup algebra C[S] is semilocal (semiprimary, right perfect, left perfect). Then every subgroup of S is finite and every nil factor of S is locally nilpotent (respectively, nilpotent, right T-nilpotent, left T-nilpotent). Let X* be the free monoid generated by a set X, and let M C X*. Elements of X* are called monomials. Denote by I the ideal generated by M in the polynomial ring R[X\. The quotient ring R[X]/I is called a monomial ring. It is isomorphic to the contracted semigroup ring RQ[S] of

Crossed

27

Products

the Rees quotient monoid S = X*/(X*MX*). If / is an ideal generated by monomials in the commutative ring R[xi..., xn], then the quotient ring R[xi... ,xn]/I is also called a monomial ring. The monograph [Villarreal (2001)] is devoted to monomial algebras. Every semigroup ring R[S] is an S-graded ring with components Rs, where s € S. US has zero 0, then the contracted semigroup ring is also graded by S with components Ro = 0 and Rs = Rs, for 0 ^ s s S. If B is a band and S = UfcgsSb is a band B of semigroups Sb, then the semigroup ring R[S] = ®beBR[Sb] is a 5-graded ring. If, in addition, R and all the Sb have identities, then R[S] is a special band-graded ring. We refer to [Passman (1977); Passman (1986)] and [Okninski (1991)] for detailed accounts on group and semigroup algebras, respectively.

3.3

Crossed Products

Let R be a ring with 1, and let S be a semigroup. A crossed product R* S of S over R consists of all finite sums Y^ses r «* with addition and multiplication defined by the distributive law and two rules xy xr

= =

r(x,y)xy r

a{x)

x

(twisting)

(3.1)

(action)

(3.2)

such that R is associative and the set S forms a multiplicative subgroup of R isomorphic to S. Every crossed product R * S is an 5-graded ring with components Ra = i?s. We refer to [Passman (1986); Passman (1989)] for further information on crossed products.

3.4

Polynomial and Skew Polynomial Rings

if xn = K 1 • • > 0} is a free commutative monoid with n generators, then 7?[Xn] is called a commutative polynomial ring in n variables. It is also denoted by R[xi,... ,xn], if it is clear from the context that commutative rings are considered. If X* is the free (noncommutative) monoid generated by X = {xi,... ,xn}, then i?[X*] is called a polynomial ring in n noncommuting variables. This ring is also denoted by R[xi,...,xn], when it is clear from the context that noncommutative rings are considered.

28

Examples of Ring

Constructions

A (left) skew derivation on a ring R is a pair (a, 6), where a is a ring endomorphism of R and 5 is a (left) a-derivation on R, that is, an additive homomorphism from R to itself such that S(ab) = a{a)6(b) + S(a)b,

(3.3)

for all a,b G R. If (a, 6) is a skew derivation on R, then the skew polynomial ring (or Ore extension) is denoted by R[S; a, 5]. It consists of all finite sums 5Zr=o rix%i where r* G # , with multiplication defined by the distributive law and the rule xr = a(r)x + S(r),

(3.4)

for all r £ R. Many quantized algebras can be expressed in terms of iterated skew polynomial rings (see [Goodearl (1992)]). The skew Laurent polynomial ring R(x; a, 6) is the set of finite sums Y^i=m.riX'> where fi G R and m , n G Z, n > m. If a = 1, then the ring R[x; 5] = R[x; 1,6} is called a skew polynomial ring of derivative type, and R(x; 6) = R(x; 1,6) is called a skew Laurent polynomial ring of derivative type. If a = 1 and 6 = 0, then we get the usual polynomial ring R[x] = R[x; 1,0] and Laurent polynomial ring R(x; 1,0). Let S be any semigroup, and let / be a homomorphism from the infinite monogenic monoid x+ = {x}+ into S. Then every skew polynomial ring of automorphism type R[x; a] = (BsesRs is an 5-graded ring with components Rs = X3/fx')=« R-xli where the sum is assumed to be equal to zero if there are no elements xl with f(xl) = s. A similar assertion is true for every skew polynomial ring of automorphism type in n commuting variables: R[xi,x2,...,xn;a]

=

R[xi;a}[x2;o-}...[xn;a],

where the automorphism a naturally extends to an automorphism of each ring R[xi,... ,Xk\o~], for k = 1 , . . . ,n — 1. 3.5

Skew Group and Semigroup Rings

Let R * S be a semigroup crossed product with twisting r and action a. If the twisting r is trivial, that is T(X, y) = 1 for all x,y G S, then R * S is denoted by R[S;cr] and is called a skew semigroup ring. If 5 is a group (or monoid), then this ring is called a skew group ring (respectively, skew monoid ring).

Twisted Group and Semigroup

29

Rings

Suppose that a is a homomorphism of a semigroup S into the endomorphism monoid (automorphism group) of the ring R. The skew semigroup ring of endomorphism (respectively, automorphism) type R[S; a] consists of all finite sums Y^i=i r»s*> w n e r e r% G -R, Sj G S, with multiplication defined by the distributive law and risi • r2s2 = ria(s1)(r2)s1s2,

(3.5)

for all r\,r2 G R, Si,s2 G S. If 5 is a group, then R[S;a] is called a skew group ring of automorphism type. Every skew semigroup ring R[S; a] is an S-graded ring with components Rs, where s G S. If S has zero 0, then the contracted semigroup ring is also graded by S with components Ro = 0 and Rs = Rs, for 0 ^ s G S. 3.6

Twisted Group and Semigroup Rings

Let R * S be a semigroup crossed product with twisting r and action a. If the action a is trivial, i.e., a{x) = x for all x, then R * S is denoted by i?T[5] and is called a twisted semigroup ring. If S is a group (or monoid), then this ring is called a twisted group ring (respectively, twisted monoid ring). Twisted group and semigroup rings are most often considered in the case where R is a field. The general definition of twisted group and semigroup rings above leads to the following concept of a cocycle. Let S be a semigroup, C a commutative semigroup, and let r : S x S —> C be a mapping. Denote by ET or ET (C, S) the set C x S with multiplication defined by the rule (c, s)(d, t) = (cdr(s, t), st) for all c,deC,s,t£

S.

(3.6)

The mapping T is called a cocycle if the following cocycle equation is satisfied: T(s,t)r(st,u)

= T(s,tu)T(t,u),

(3.7)

for all s, t, u € S. The cocycle equation implies the associativity of ET. If C is cancellative, then the set ET is a semigroup if and only if T is a cocycle. The semigroup ET is called the extension semigroup. Thus, if R is a field, then the associativity of the twisted semigroup ring RT \S] is equivalent to r being a cocycle. It is worth noting that the cocycle equation arises in

30

Examples of Ring

Constructions

the topology of surfaces, in quantum mechanics, in group cohomology, in combinatorial designs, and constructing error-correcting codes (see [Baliga and Horadam (1995)], [de Launey et al. (2000)], [Horadam (1996)], [Horadam (2000)], [Horadam and de Launey (1993)], [Horadam and de Launey (1995)], [Horadam and Perera (1997)] and [Horadam and Udaya (2000)] for details).

3.7

Power and Skew Power Series Rings

If [a, 5) is a skew derivation on R, then the ring R[[x; a, 6]] of skew power series of R is the set

R[[x; a, 6}} = 0

'n

G R

and the ring of skew Laurent series of R is the set

R((x,a,S))

=< ^

nxl | n e R,m £ Z

I i>m

with multiplication defined by (3.4). If a = 1, then the ring i?[[x;£]] = R[[x; 1,5]] is called a skew power series ring of derivative type, and the ring R((x;5)) = R((x;l,5)) is called a skew Laurent series ring of derivative type. If 5 = 0, then the ring ^[[ZJCT]] = R[[x; a, 0]] is called a skew power series ring of endomorphism type, and R((x;5)) = R((x;l,S)) is called a skew Laurent series ring of endomorphism type. If, in addition, a is an automorphism, then these rings are called the ring of power and Laurent series of automorphism type. If a = 1 and (5 = 0, then we get the usual power series ring R[[x]] = R[[x; 1,0]] and R((x; 1,0)). Let S be any semigroup, and let / be a homomorphism from the infinite monogenic monoid x+ into S. Then every skew power series ring of automorphism type i?[[x;cr]] = (BsesRs is a n 5-graded ring with components Rs = ^2f(xi)=s Rxl, where the sum is assumed to be equal to zero if there are no elements xl with f(xl) = s. This transfers to all skew power series rings of automorphism type in commuting n variables: R[[xi,x2,

...,x„;o-]] = R[[xi; 0 is a sequence of vertices Vo, v\,..., vn such that (i>j_i, Vi) G E, for all i = 1 , . . . ,n. In particular, a single vertex is a path of length 0. Denote by P(D) the set of all paths of D, and introduce a multiplication on P°(D) = P(D) U {0} by putting, for any two paths u = ui,...,um and v = v1,...,vn, _

( M l , . . . ,Um,V2,..

10

.,Vn

if i um = vi otherwise.

(3.8)

The path algebra of D over F is denoted by FD and is defined as the contracted semigroup ring F0[P°(D)]. The study of path algebras, in particular, path algebras of quivers and their homomorphic images, plays essential role in the representation theory.

3.9

Matrix Rings and Generalized Matrix Rings

Let R be a ring, and let I be any set. The set of all matrices (upper triangular matrices) with entries in R and rows and columns indexed by the elements of I will be denoted by Mi(R) (respectively, Ui(R)). If |7| = n is a positive integer, then this set is denoted by Mn(R) (respectively, Un(R)), and it is known that it forms a ring with respect to the usual matrix multiplication. This ring is called the full matrix ring (upper triangular matrix ring) over R. A matrix is row (column) finite if each of its rows (columns) has a finite number of nonzero entries. The set of all row (column) finite matrices of MT(R) forms a ring denoted by MJ rf) (i?) (respectively, M$o!)(R)) and

32

Examples of Ring

Constructions

called the ring of row (column) finite matrices over R. The intersection M 7 (rcf) (i?) = Mff){R)nMff)(R) is an ideal in both M^(R) and M\C1)(R). It is called the ring of row and column finite matrices over R. A matrix is row (column) bounded if it has a finite number of nonzero columns (rows). The set of all row (column) bounded matrices of Mi(R) is a ring. It is denoted by MJ rb) (i?) (respectively, MJ cb) (i2)) and called the ring of row (column) bounded matrices over R. The intersection M\ (R) — Mf (R) D Mj (R) is the ring of matrices with finite numbers of nonzero entries. A ring R = ®™j=1Rij is called a generalized matrix ring if

RijRk,i

c

Ri,e 0

if j = k; otherwise.

It is said to be a generalized upper (lower) triangular matrix ring if Rij = 0 for all 1 < j < i < n (respectively, 1 < i < j < n). A generalized lower triangular 2 x 2 matrix ring was used to construct examples of nonsingular rings (see [Goodearl (1976)]). Generalized row finite matrix rings are defined in a similar fashion, as well as column finite, row and column finite, row bounded, column bounded matrix rings. All of these rings are graded by Brandt semigroups, finite groups, and rectangular bands. To illustrate the standard grading methods, consider the ring K = Mn(R). Recall that the Brandt semigroup Bn consists of 0 and all standard matrix units. Putting Ko = 0 and Ks = Rs for 0 ^ s £ Bn, we see that K = © s £ B Ks is graded by the Brandt semigroup Bn. Take any group G of order n. In order to index the rows and columns of K by the elements of G, we introduce a one-to-one mapping from { 1 , . . . ,n} to G. The z-th row and i-th column are indexed by 4>(i). For g e G, put Lg = E ^ - i ^ O f f ^ . r T h e n K = ®geGL9 i s a G-graded ring. If H = {(i,j) | 1 < i,j < n} is a rectangular band, then K = ®heH Ph is an i7-graded ring, where Ph = Reitj. There are also other ways of grading Mn(R) by rectangular bands, as mentioned in Section 2.2.

Triangular Matrix Representations

3.10

33

Triangular Matrix Representations

If R is a ring with 1 graded by the Brandt semigroup Bn, and if i? 0 and all components Rij = Rei . are equal to zero for all i > j , then R is also called a generalized triangular matrix representation of R, since it can be viewed in matrix notation. An internal characterization of generalized triangular matrix representations of rings in terms of sets of left triangulating idempotents has been obtained, and a number of important properties of these rings have been described in [Birkenmeier et al. (2000)]. Diverse applications of generalized triangular matrix representations occur in operator theory, quasitriangular Hopf algebras, and various Lie algebras (Kac-Moody, Lie, Virasoro, and Heisenberg). A variety of conditions have been used in order to obtain generalized triangular matrix representations of algebras (see [Birkenmeier et al. (2000)] for references). Triangular matrix representations are also graded by groups, Brandt semigroups, and rectangular bands, as illustrated in Section 3.9.

3.11

Morita Contexts

If R = ®seB2Rs is a contracted i?2-graded ring, then Ren

) -*^ei2 : *^e2\ > •*^e22j

is called a Morita context. Conversely, if [P, U, V, Q] is a Morita context, and we put Reil = P, Rei2 = U, Re21 = V, Re22 = Q, then R = ®s^B2Rs is a contracted 5 2 -graded ring. Thus the definition of a Morita context is equivalent to the definition of a contracted i?2-graded ring. Morita contexts play crucial roles in the investigations of categories of modules over rings.

3.12

Rees Matrix Rings

Let R be a ring, I and A two sets, and let P be a A x /-matrix with entries in R. A Rees matrix ring over R with sandwich matrix P is the set M(R; I, A; P) consisting of all J x A matrices over R, with finite numbers of nonzero entries, equipped with the usual addition and multiplication • defined by A • B = APB, for A,B £ M(R; I, A; P). Row finite (column finite, row and column finite, row bounded, column bounded) Rees matrix

34

Examples of Ring

Constructions

rings are defined similarly. These rings are graded by groups, rectangular bands, and Brandt semigroups, see Section 3.9.

3.13

Smash Products

Let G be a finite group, and let R be a G-graded ring. Then the smash product R#Z[G*] of R and Z[G*] is the collection of all matrices M with rows and columns indexed by the elements of G and such that Mg>h € Rgh-\, for all g,h £ G. The ring R embeds in i?#Z[G*] by the mapping •K such that n(r)gth = rgh-i, for all g,h £ G. The whole smash product is generated by R and the standard matrix units pg = e 5;5 , for g € G. If R is an algebra with 1 over a field F, then we can also define the smash product R#F[G*} of R and G as the free left and right i?-module with a basis {pg \ g £ G} consisting of pairwise orthogonal idempotents whose sum is 1 and with multiplication given by the rule (apg)(bph) = abgh-iph.

(3.9)

Each smash product is a generalized matrix ring, and therefore it is graded by a group, a rectangular band, and a Brandt semigroup.

3.14

Structural Matrix Rings

Let R be a ring with 1, and let D = (V, E) be a directed graph with the set V = { 1 , 2 , . . . , n) of vertices and the set E of edges without multiple edges but possibly with loops. Edges of D correspond to the standard matrix units of the algebra Mn(R) of all (n x n)-matrices over F. Namely, for (i,j) £ E C V x V, let e (»>j) = e*>J — eiJ ^ e the standard matrix unit. Denote by MD(F) = ®weEFew the set of all matrices with nonzero entries corresponding to the edges of the graph D, and zeros in all entries for which there are no edges in D. It is well known and easy to verify that MD{F) is a subalgebra of Mn(F) if and only if D satisfies the following property (x,y),(y,z)

€ E =*• (x,z) £ E,

(3.10)

Incidence

Algebras

35

for all x,y,z € V. In this case the MD(F) is called a structural matrix ring. Structural matrix rings have been investigated by a number of authors, and many interesting results have been obtained (see, for example, [Dascalescu et al. (1999)], [Dascalescu and van Wyk (1996)], [Green and van Wyk (1989)], [van Wyk (1996)], [van Wyk (1996)], [van Wyk (1999)], [Veldsman (1996)]). Note that every structural matrix ring can be regarded as a semigroup ring. A graph D = (V, E) defines a structural matrix ring if and only if the set SD =

{0}U{eij\(i,j)GE}

forms a semigroup, and both of these properties are equivalent to condition (3.10). Then the structural matrix ring MD(F) is isomorphic to the contracted semigroup ring -POISD]The following definitions are used in the investigations of structural matrix rings. Let D = (V, E) be a graph. The in-degree and out-degree of a vertex v &V are defined by indeg(w) = \{w e V \ (w,v) € E}\, outdeg(u) = \{w G V | (v,w) € E}\. A vertex of D is said to be a source if indeg(u) = 0 and outdeg(v) > 0. A clique of D is a maximal complete subgraph of D. A graph is acyclic if it has no directed cycles.

3.15

Incidence Algebras

Let R be a ring, and let X be a partially ordered set. For x, y e X, the interval between x and y is the set [x,y] = {z \ x < z < y}. A partially ordered set is locally finite if every interval [x, y] is finite, for all x, y S X. The set I(X,R)

= {f:XxX-+R\

f(x,y)

= 0 if x £ y}

with operations

(f + g)(x,y) =

f(x,y)+g(x,y),

36

Examples of Ring

fg(x,y)=

^2

Constructions

f(x,z)g(z,y)

and

z£[x,y]

(rf)(x,y)

=

rf(x,y),

for all f,g £ I(X,R), r £ R, x,y,z € X, is called the incidence algebra of X over R. We refer to [Spiegel and O'Donnell (1997)] for a detailed exposition on incidence algebras. The following definition is more general. Let R be a ring, X a finite set, and let p be any reflexive relation on X. The incidence ring I(X,p,R) of X with coefficients in R is the free left .R-module with basis consisting of ordered pairs {(x,y) \ xpy}, where multiplication is defined by the distributive law and si \ ( (x:w) •v (z, w) = { \ ' ' ' \ 0

i

v(x,,yjy)

if y = z and xpw, F , . otherwise.

A relation p is said to be balanced if, for all x\,xi,xz,x\ X2pX3,

X3PX4,

,_ , , .

v (3.11) ;

€ X with x\px2,

X1PX4, X\pXz

•& X2PX4.

It is proved in [Abrams (1994)] that I(X, p, R) is an associative ring if and only if p is balanced. If X be a finite set with a preorder 1. Denote by E the subsemigroup generated by e. We claim that P C E. Indeed, suppose that fk £ P for some k > 1. Then k > m by choice of p. Let 1 be the identity of the group G. Since em £ P C G, there is a u in G such that emu = 1. Also, e m l = e m and / f c l = fk because e " \ / f e € G; the former implies that ekl = ek since k > m. But / e = e 2 implies ekem = fkem. k k k m k m k k fe Hence f = f l = f e u = e e u = e l = e , so that / £ E. If e is not periodic, then E is an infinite cyclic semigroup and so P C E is isomorphic to a subsemigroup of Z. By Theorem 4.7, J{RG) = <J(Rp) is homogeneous. If e is periodic, then ife = Rp = F[P] and P = G C\ E is a finite group (since it is finite and cancellative). Then J(RG) = 0 by Theorem 3.1, and therefore J{RG) is certainly homogeneous. So J{RQ) is homogeneous for all subgroups G of S. Since S is a JGH-semigroup, it follows that J{R) is homogeneous. Put d = e — f. For any u £ V the equality ei> = fv holds, implying dR = 0 and d £ J(R). The homogeneity of J{R) implies e e J"(.R). This is not possible since R — F[V] and e has augmentation 1. • Lemma 4.12 Let S be a JGH-semigroup. If u,v £ S, u ^ v, then for any w £ S, the equalities uw = vw and wu — wv are equivalent. Proof. Suppose to the contrary that uw = vw but wu ^ wv. Putting e = wu and f = wv we get ef = f2 and fe = e 2 , a contradiction to Lemma 4.11. • Proposition 4.2

Let S be a JGH-semigroup.

Then S is cancellative.

Homogeneous

Radicals

55

Proof. Suppose that S is not cancellative. Then there exist u,v,w £ S such that u ^ v but uw = vw. (By 4.12, uw = vw is equivalent to urn = wv.) We will construct an S-graded ring R for which J{RG) is homogeneous for all subgroups G of S but J(R) is not homogeneous. Denote by M the ring of 2 x 2 matrices over the complex numbers C. Let ei2 = I

I be the standard matrix unit and put N = Cei2- Then

JV2 = 0. Let W be the ideal of S generated by w. Define a subring R of the semigroup ring M[S] by R — NS + MW; this is a subring, because W is an ideal of S. Since R is a homogeneous subring of M[S] it inherits the natural 5-gradation from M[5]. Let G be a subgroup of S. Since W is an ideal of S, two cases are possible: G n W = 0 or G C W. In the former case, RG C iVS so i? G 2 = 0 and therefore J(RG) — RG- In the latter case, RG is the group ring M[G], which is isomorphic to the ring of 2 x 2 matrices over the group ring C[G]. But J(C[G\) = 0 (see [Passman (1977)]) so J(RG) = J(M[G}) = 0 (by [Karpilovsky (1991)], Proposition 6.13). In either case, J{RG) is homogeneous. Consider the element d = e ^ w — v) of the ring R. Take any s G W, say s = awb where a, b £ S1. Since uwb = vwb, 4.12 yields wbu = wbv; then su = awbu = awbv = sv and us — vs again by 4.12. Therefore dRs = 0 for s£W. If s£S\W then dRs C e12NS = 0. So dfl = 0 and d £ J(R). If J{R) were homogeneous, we would have e^u £ J{R). But the augmentation of e\2u (as an element of the semigroup ring Mf.!?]) is ei2, a contradiction, since the image of R under the augmentation map is M and

J{M) = 0 .

•

Theorem 4.12 Let S be a cancellative semigroup, and let R be an S-graded PI-algebra over a field of characteristic zero. Then the Jacobson radical of R is homogeneous. Easy examples of group algebras of finite groups show that the restriction on characteristic cannot be removed from Theorem 4.12. Theorem 4.13 Let S be a u.p.-semigroup, and let R be an S-graded ring such that at least one of the following conditions is satisfied: (i) all nil subsemigroups of H(R/J'gI(R)) are locally nilpotent; (ii) every nil subsemigroup of every right primitive homomorphic image of R is locally nilpotent;

56

The Jacobson

Radical

(iii) for every minimal prime ideal P of R, the ring R/P or embeds into a matrix ring over a division ring.

is a domain

Then the Jacobson radical of R is homogeneous. The class of rings satisfying (i) contains all rings R such that in all homomorphic images of R all multiplicative nil subsemigroups are locally nilpotent. This applies to all Pi-rings, left or right Noetherian rings and, more generally, all rings with left or right Krull dimension (see [McConnell and Robson (1987)], 6.3.5,13.4.2). The class of rings satisfying (ii) contains, beyond the classes mentioned above, all semilocal rings. Condition (iii) concerns all rings which are 'nice' modulo the Baer radical. In particular, this covers the case where R/B(R) is a right Goldie ring. If J C J(R), then J(R) is homogeneous if and only if J(R/I) is homogeneous. If / C B(R), then B(R) is homogeneous if and only if B(R/I) is homogeneous. Proposition 4.3 Let S be a u.p.-monoid with identity e, R an S-graded ring, r a rigid element of R, and let M be the multiplicative semigroup generated by H(r). Then (i) if r 0 Re and M^rM1 consists of quasiregular elements, then M is nilpotent; (ii) ifH(R)1rH(R)1 consists of quasiregular elements, thenre £ J(Re), and if, additionally, r 0 Re, then re belongs to the nilradical of Re. The following lemma allows us to consider rings and semigroups with identities. Lemma 4.13 (i) Let S be a u.p.- or t.u.p.-semigroup without identity element. Then the semigroup Se with identity e adjoined is also u.p. or t.u.p., respectively. (ii) Let R = (BsesRs be an S-graded ring. Denote by Re the subring generated in R} by 1. Then R1 = © ^ e s 1 ^ *s S1-graded. Proof. The assertion (ii) is obvious. In (i) we only consider the case where S is a t.u.p.-semigroup, since the proof for u.p.-semigroups is similar. Take two nonempty subsets X,Y £ Se with \X\ + \Y\ > 2, and any elements a,b £ S. The sets aX and Yb are contained in S, and \aX\ + \Yb\ > 2, because Se is cancellative. Therefore there exist distinct elements u', v' uniquely expressed in the form u' — axyb, v' = aztb, where ax, az £

Homogeneous

Radicals

57

aX, yb,tb £ Yb. Put u — xy, v = zt. By the cancellativity of S these representations of u and v as products of elements from X and Y are unique. Thus Se is a t.u.p.-semigroup. • Lemma 4.14 Let S be a cancellative semigroup, R an S-graded ring, r a rigid element of R, and let M be the multiplicative semigroup generated by H(r) in R. If M contains 0, then M is nilpotent. Proof. Suppose that M contains 0. This means that y\... ym = 0 for some 2/i, • • •, ym e H(r). Choose m and yi,...,ym such that q = yx... y m _ x ^ 0, and consider the product z = qr. Since q is homogeneous, z is also a rigid element of R. Given that S is cancellative, we see that qym is a homogeneous component of z. Therefore qym = 0 implies qrs = 0 for all s £ S; whence z — 0 and, moreover, yi... ym-iH(r) = 0. Similarly, considering 2/i • ..ym-2rH(r) = 0, we get yx.. .y m _ 2 (i?(r) 2 ) = 0. Repeating this m times, we conclude (H(r))m = 0. Thus M is nilpotent. • Lemma 4.15 Let S be a cancellative semigroup, R an S-graded ring, I the ideal of nonunits of S and G = S\I (if S has no identity, then S = I). Assume that x + y = yx for some x 6 Ri,y £ R. Then y € K, where K is the subring generated by H(x). Proof. We will show that yg e K, for each g e I. The case where yg = 0 is trivial, and so we assume g £ supp(y). First, note that y £ Rj, because Rj is an ideal of R. If two elements s and t of I generate the same right ideal, then s = t. Therefore there exists a maximum positive integer n such that supp(^i)/ 1 D ... D supp(2:„)/ 1 , for some z\,...,zn £ H(y), where zn = yg. We call n the depth of yg. We proceed by induction on the depth of yg. Assume first that the depth of yg is 1. This means that gl1 is maximal in the set of right ideals of I generated by the elements of supp(y). Hence g = st implies ys = 0, and so yg £ H(x) C K. Next, assume that the depth of yg is n > 1. Since x + y — yx, we get Vg ~ Y^st=gysxt — xg. If st = g and ys ^ 0, then the depth of y3 is less than n, and by the inductive assumption yg £ K. It follows that yg £ K, as claimed. • Now we can prove Proposition 4.3. Proof. By Lemma 4.13, we may assume that R has an identity 1. Every u.p.-group is t.u.p. (see [Okniiiski (1991)], Chapter 10), and so the group

58

The Jacobson

Radical

of units G of S is a t.u.p.-group. It is routine to verify that I = S \ G is an ideal of S. Assume that r $ Re and M1rM1 consists of quasiregular elements. By Lemma 4.14, in order to show that M is nilpotent it suffices to prove that OGM. First, consider the case where r £ RG- In view of [Karpilovsky (1991)] Corollary 22.9, all elements of M1rM1 are quasiregular in RG- Since G is t.u.p., we can follow the argument used in [Jespers et al. (1982)]. Replacing r by rgrrh for some g,h £ supp(r), without loss of generality we may assume that all elements of M1rM1 are quasiregular, and that e ^ supp(r). Then |1 — r\ > 1 and 1 — r is a unit of RG- Therefore (1 — r)b = 1 for some b £ RG, and so ck(l — r)b = ck for any c £ H(r) and any k. Choose c £ H(r), k > 1, and b such that |6| is minimal among the lengths of all elements b satisfying ck{\ —r)b = ck for some c, k. lib = 0, then 0 = ck £ M and we are done. Suppose b ^ 0. Since S is t.u.p. and |1 — r\ > 1, we can find an element w £ S,w ^ supp(ck), uniquely expressed in the form w = uv, where u £ supp(ck(l — r)), v £ supp(b). Then ck{\ — r)b = ck implies [c^l — r)]ubv = 0. Since r is rigid and c £ H(r), it follows easily that ck+1{l-r)bv = 0. Therefore c fe+1 (l -r)(b-bv) = 0, contradicting the choice of b, and proving the claim. Second, suppose that r g- RG- Then there exists h £ I such that r^ ^ 0. Given that / is an ideal of S, we get b = rhr £ Rj. Obviously, b inherits the hypothesis imposed on r, and so we may assume that from the very beginning r £ Rj. Suppose that the semigroup M does not contain 0. Denote by T the subsemigroup generated in S by supp(r). Then, for every t £ T, we have

M n Rt ± 0. Suppose that T is not a right Ore semigroup. Let p be the left reversive congruence on T defined in [Okninski (1993a)]. Then there exists t £ T such that the set isupp(r) is /9-separated in the sense of [Okninski (1994)] and (h, 1) ^ p for every h £ isupp(r). Choose a nonzero element b £ M n Rt. We know that x = br ^ 0 and x + y = xy = yx for some y £ Rj. By Lemma 4.15, y £ RTThe left cancellativity of p implies that x' + y' — x'y' = y'x' for every p-class in T and the corresponding p-components x', y' of x, y. Let X be the semigroup generated by the support of x. Replacing x by any cxd, where c,d £ M, we see that the element cx'd has a quasi-inverse u £ Ri- Again,

Homogeneous

59

Radicals

we know that u £ Rx- Therefore x' and Rx inherit the hypotheses on x and R. Proceeding in this way we eventually come to an element z whose support is in a single /o^-class, where pz is the left reversive congruence on the semigroup Z generated by the support of z. Prom [Okninski (1993a)], Lemma 3, we know that Z is a right Ore semigroup. Hence Z is a t.u.p.-semigroup (see [Okninski (1991)], Theorem 10.6). Since z inherits the hypotheses on r, we can apply the first paragraph of the proof to see that 0 is contained in the semigroup generated by the components of z. But z £ M, so this contradicts the assumption that 0 £" M. If T is a right Ore semigroup, then it is t.u.p., and we get a contradiction again. It follows that 0 £ M, which completes the proof of (i). Further, we show that (ii) easily follows from (i). Indeed, assume that H(R)1rH(R)1 consists of quasiregular elements. If r £ -Re, then RerRe C H(R)1rH(R)x consists of quasiregular elements. By [Karpilovsky (1991)], Corollary 22.9, all these elements are quasiregular in Re, as well. Therefore re £ J{Re). Assume that, in addition, r 0 Re. Since all elements in H(R)lrH(R)1 are quasiregular, clearly (i) applies to every nonzero xry, where x, y £ H(R)1. Therefore L = L U s ^ C R ) 1 ^ . ^ ) 1 is a nil ideal of H(R). In particular, re generates a nilideal of Re. This completes the proof. • L e m m a 4.16 Let R be an S-graded Pi-ring, and let T be a multiplicative subsemigroup of R. If T does not contain zero, then supp(T) is a permutational subsemigroup of S. Proof. Let H = supp(T). Every Pi-ring (or Pi-algebra) satisfies a multilinear identity. Let n be the degree of a multilinear identity (1.5) satisfied ini?. Take any elements t 1 , i 2 , • • • ,*n in T. Suppose that ti £ R^, for i — 1 , . . . , n. Applying (1.5) to the elements ti,...,tn we get h ...tn

£ Rht...h„ n

22

R

hal...han-

Given that T does not contain 0, it follows that t i . . . tn ^ 0. Therefore 0^t1...tn€Rhl...hnnRha,b '• Ra -^ B-b such that xy = 4>a,ab{x)4>b,ab{y) for all a,b € S, x e Ra, y G Rb, and b,c(a,b{x)) = <j>a,c(x) f°r a^ a -< b -< c. Then J{R) = {reR\

Y,

Mr

a) € J{Rb)

for all

beS}.

a with ab=b

This theorem is proved in [Bell et al. (1996)] even for a large class of socalled directed hereditary radicals (and so it is also true for Levitzki radical). Theorem 4.16 covers all commutative combinatorial periodic semigroups and all finite bands. A description of prime ideals of the rings satisfying the hypotheses of Theorem 4.16 is given in [Bell et al. (1996)]. Theorem 4.17 ([Kelarev (1990b)]) The Baer radical is determined by the idempotents of a semigroup S if and only if S is a locally finite combinatorial semigroup and every Rees factor of S has a nilpotent or minimal ideal. The main theorem of [Kelarev (1991a)] divides all bands into 18 classes, and, for bands B of each class, describes all radicals determined by the components of B-graded rings. Let us include only one corollary. Corollary 4.2 ([Kelarev (1991a)]) Let B be a band, and let Y be the largest semilattice homomorphic image of B. Then the following conditions are equivalent:

66

The Jacobson

Radical

(i) for every B-graded ring R = ©bgB-Rfc; the Baer radical B(R) is equal to the largest ideal I such that fs3\(x) £ B(RS), for all x € I,seS; (ii) S satisfies the descending chain condition for idempotents. The next interesting question is that of when there exists a nice relation between the radical of the whole ring R and those of homogeneous components Re, where e = e 2 . The investigation of radicals which have a good structure in graded rings was initiated by Gardner [Gardner (1975)]. Recall that a mapping g is called a radical if, for every ring R with ideal / , (i) g(R) is an ideal of R;

(ii) g(R/g(R)) = 0; (iii) g(R/I)2g(R)+I. All of these properties hold for the Jacobson, Baer, and Levitzki radicals. A radical g is strict if g(T) C g(R) for every ring R with a subring T. All strict radicals were characterized by Stewart [Stewart (1973)]. A radical is said to be weakly hereditary if g(I) 2 g(R)I + Ig(R) for each ring R and every ideal / of R. T h e o r e m 4.18 ([Kelarev (1989b)], [Kelarev (1992c)]) Let S be a nontrivial semilattice, and let g be a radical. Then the following conditions are equivalent: (i) (ii) (iii) (iv)

g(®sesRs) = ®SGSQ(RS) for every S-graded ring R; g((BsesRs) C ®sesg(Rs) for every S-graded ring R; g(Re) = Re n Q(R) for any S-graded ring R and any e £ S; g is strict and weakly hereditary.

Theorem 4.19 ([Kelarev (1992c)]) Let S be a nontrivial semilattice, and let g be a radical. Then the following conditions are equivalent: (i) (ii) (iii) (iv)

g is homogeneous in every S-graded ring; g(®sesRs) 2 ®s€SQ(Rs) for every S-graded ring R; if R = (BstsRs is g-semisimple, then all Rs are g-semisimple; g is strict.

T h e o r e m 4.20 ([Kelarev (1995b)]) Let S be a semigroup with zero, E the ideal of S generated by all idempotents. Then the following are equivalent: (i) J {Re) = J{R)C\Re for each contracted S-graded ring R = ©Sgs-Rs and every idempotent e in S;

Nilness and

Nilpotency

67

(ii) the largest nil ideal N of E is locally nilpotent, and if E ^ 0, then E/N is a 0-direct union of inverse completely 0-simple semigroups with locally finite maximal subgroups. Let S be a semigroup with zero. A radical g is said to be invariant in S-graded rings if and only if Rxg(Re)Ry C g(Rf) for every contracted S-graded ring R = (BstsRs, where e, / are any nonzero idempotents of S and x, y are any elements of S such that xey = / . Since this definition involves only nonzero idempotents, it applies to graded rings and contracted graded rings in exactly the same way. Sands [Sands (1989)] proved that a radical is invariant in a group-graded ring or a Morita context if and only if it is normal. Normal radicals were described by Sands [Sands (1975)]: A radical is normal if and only if it is left strong and principally left hereditary. A radical is said to be left strong if every radical left ideal of each ring R is contained in the radical of R. A radical is principally left hereditary if, for every radical ring R and element a £ R, the left ideal Ra is also radical. For any ring R, the ring with zero multiplication and the same additive group as R will be denoted by R+. A radical g is called an A-radical if g(R) = g(R+) for every R. Theorem 4.21 ([Gardner and Kelarev (1997)]) Let g be a radical, S a semigroup, E the ideal generated by all idempotents in S, and let N be the union of all ideals of S which are contained in E, but do not contain nonzero idempotents. Then the radical g is invariant in S-graded rings if and only if at least one of the following conditions holds: (i) g is an A-radical; (ii) the quotient semigroup E/N is a semilattice of height < 2; (iii) all principal factors of the quotient semigroup E/N are singletons and g is strict and weakly hereditary; (iv) the radical g is normal and the quotient semigroup E/N is an inverse semigroup in which every nonzero idem/potent is primitive.

4.6

Nilness and Nilpotency

A number of classical theorems of ring theory deal with nilness and nilpotency of the Jacobson radical of various ring constructions (see [Karpilovsky

68

The Jacobson

Radical

(1991)], [Rowen (1991)]). The following new results have been obtained recently. Theorem 4.22 ([Clase and Jespers (1994)]) Let S be a finite (locally finite) semigroup, and let R = © geC ;i? g be an S-graded ring. If the radicals J{Re) are nilpotent (locally nilpotent) for all e £ E(S), then J{R) is nilpotent (respectively, locally nilpotent), too. Theorem 4.23 ([Kelarev and Okniriski (1995)]) Let G be a group with identity e, and let R = (BgeGRg be a G-graded Pi-ring. If J{Re) is nil, then J{R) is nil, too. Theorem 4.24 ([Kelarev (1993d)]) Let G be a non-torsion group with identity e. Then there exists a strongly G-graded ring Q such that J(Qe) = B(Qe) butJ{Q)^C{Q). The theorem above answered a question asked in [Puczylowski (1993)]. Theorem 4.25 ([Kelarev (1993d)]) Let S be a commutative The following are equivalent:

semigroup.

(i) for every strongly S-graded ring R = ®sesRs, if all the Jacobson radicals J{Re) are nilpotent for all e £ E(S), then the radical J{R) is nilpotent, too; (ii) S has an ideal I such that S/I is nilpotent and I is a finite union of groups with finite torsion parts. Theorem 4.26 ([Kelarev (1993d)]) Let S be a linear epigroup over any field, and let R = ®s&sRs be a strongly S-graded algebra over a field of characteristic zero. If all radicals J{Re) are nilpotent for e £ E(S), then J{R) is nilpotent, too. Lemma 4.18 Let R be a G-graded PI-ring, I a homogeneous ideal of R contained in J{R). If Ie is nil, then I is nil. Proof. Take any element r in H(I). Since I is homogeneous, r £ I. Let r £ J(R) n Rg, where g £ G. If g is a periodic element, then there exists a positive integer n such that rn £ lDRe — Ie, and so r is nilpotent. Further, assume that g is not periodic. Denote by T the infinite cyclic group generated in G by g. Lemma 2.2 shows that r £ J(RT), and therefore r is nilpotent again in view of [Karpilovsky (1991)], Theorem 32.5. Thus H(I) is a multiplicative nil subsemigroup of R. Since R satisfies a polynomial

Nilness and

Nilpotency

69

identity, it follows from [Rowen (1980)], Theorem 1.6.36, that / is nil, as required. • Lemma 4.19 Let G be a permutational group, R a G-graded Pi-ring. J(Re) is nil, then J{R) is nil.

If

Proof. By Theorem 1.18, G is finite-by-abelian-by-finite. Take any r £ J (Re)- Denote by S the subgroup generated in G by the support of r. It is easily seen that S is also finite-by-abelian-by-finite. Lemma 2.2 implies r € J{Rs), and so without loss of generality we may assume that G is finitely generated itself. By Theorem 1.18, G is abelian-by-finite, that is G has an abelian normal subgroup A of finite index. Then A is finitely generated, too (see [Kargapolov and Merzljakov (1979)]). Therefore G contains a torsionfree abelian subgroup T of finite index. By Lemma 2.1, R is graded by the finite group G/T with the identity component RT- Therefore [Okniiiski (1986a)], Lemma 1.1(1), shows that it suffices to prove that J{RT) is nil. However, J(RT) is homogeneous by Theorem 4.7, because T is torsion-free abelian. Lemma 4.18 completes the proof. • Lemma 4.20 Let G be a permutational group, R a G-graded Pi-algebra over a field of characteristic zero. If J{Re) is nil, then J(R) is homogeneous. Proof. We verify that H{J(R)) consists of nilpotent elements. Then [Rowen (1980)], Theorem 1.6.36, will show that H(J(R)) generates a homogeneous nil ideal I in R, and so J(R) = I is homogeneous. • Pick any r G J{R) and g e supp(r). We claim that rg is nilpotent. As in the beginning of the proof of Lemma 4.19, we may assume that G has a torsion-free abelian subgroup T of finite index. If we look at the natural G/T-gradation of R and apply Lemma 2.1, Theorem 4.7 and the fact that our field has characteristic zero, then we conclude that J{R) is G/T-homogeneous. We may assume that the whole supp(r) is contained in one T-coset of G (otherwise we would pass to the G/T-homogeneous summand of r corresponding to the coset containing g). Since G/T is finite, there exists a positive integer n such that rr™ £ RT. Given that J(Re) is nil, [Karpilovsky (1991)], Theorem 32.5, implies that all the homogeneous summands of rr™ are nilpotent. Therefore rg is nilpotent, as required. Now we can prove Theorem 4.11.

70

The Jacobson

Radical

Proof. By Lemma 4.18, the largest homogeneous ideal I of R contained in J(R) is nil. Obviously, R/I is a G-graded ring, and J(R/I) = J(R)/I. Therefore it suffices to prove Theorem 4.11 for R/I. To simplify the notation we may assume from the very beginning that 1 = 0. Suppose that J{R) ^ 0. Choose a nonzero element r with a minimal length in J(R). Denote by T and S the subgroup and, respectively, subsemigroup generated in G by supp(r). Let M = M{r) be the multiplicative subsemigroup generated in R by H(r). We claim that H (r) consists of nilpotent elements. If S is permutational, then the group T is also permutational by Theorem 1.18, and so all elements in H(r) are nilpotent in view of Lemmas 4.19 and 4.20. Further, consider the case where S is not permutational. Let n be the degree of a multilinear identity (1.5) of R. There exist elements sx,...,sn in S such that Si... sn ^ s CT i... sCT„, for all 1 ^ a € Sn. Clearly, there exist xi,..., xn € M such that Xi £ RSi for alii = 1 , . . . , n. Applying (1.5) to the elements x\,... ,xn we get

xi...xneRSl...Snn

^2

Rs„i...s„n,

(4-1)

whence xi...xn = 0. It follows that yi... ym = 0 for some yi,...,ym € H(r). Then we can choose m and 2/1,..., ym such that yi... ym-i ¥" 0Now look at the product y\... ym-ir. It also belongs to J(R) but has fewer homogeneous summands than r. Hence j / i . . . ym-ir = 0 by the choice of r. Since G is a group, we get y\... ym-\H{r) = 0. Further, we can look at yi... ym-2'>'H(r) = 0 and infer j / i . . . ym-2{H(r)2) = 0. Reasoning like this m times, we conclude (H(r))m = 0. In particular, all elements in H{r) are again nilpotent. Denote by L the ideal generated in H(R) by H{r). Each nonzero element of L is a homogeneous summand of a certain element of positive minimal length in J{R). It follows from what we have proved, that L is a nil ideal of H{R). Hence L is locally nilpotent by [Rowen (1980)], Theorem 1.6.36. Therefore L generates a nil subalgebra K in R. Evidently, K is a homogeneous ideal of R, and so K C I, contradicting the fact that 7 = 0. Thus J{R) = I, and so J{R) is a homogeneous nil ideal of R. D Corollary 4.3 Let G be a group with identity e, and let R = ©sgG-R9 be a strongly G-graded PI-algebra over a field of characteristic zero. IfJ{Re)

Nilness and Nilpotency

71

is nilpotent, then J(R) is nilpotent. Proof. By Theorem 4.11, we get J(R) = ®g 1 by [Rowen (1991)], Proposition 2.6.30. Since M is, evidently, closed under multiplication by elements of H(R), it follows that M generates a nilpotent ideal in R. This contradicts the primeness of R and completes the proof. •

Chapter 5

Groups of Units

If R is a ring with identity element 1, then denote by U(R) the group of units of R, i.e., the set of left and right invertible elements. Every ring is a monoid with respect to the circle composition o defined by xoy = x+y—xy. The group of units of this adjoint monoid (R, o) is called the adjoint group of R and is denoted by V(R). If R has an identity element, then it is easy to verify that x £ U(R) & 1 — x £ V(R). Moreover, the group of units is isomorphic to the adjoint group. Theorem 5.1 ([Kelarev (1994d)]) Let R = (BbeBRb be a special bandgraded ring, and let x £ R. Then x £ V{R) if and only if ltaJr-lb £ V{Rb) for each b £ subs(:r). By supp B (r) we denote the set of all b £ B such that rj, ^= 0. Let subs(r) be the subsemigroup generated in B by supp B (r). Since every band is locally finite, subs(r) is a finite set. The next lemma was obtained in [Chick and Gardner (1987)] and in [Teply et al. (1980)]. Lemma 5.1 Let C be a semilattice, R = ®cecRc a semilattice graded ring, c £ C, and let f : R —> R- be the mapping defined by f(r) = r-~ for r £ R. Then f is a homomorphism of R onto R-. Now let C be a semilattice, R — (BcecRc a semilattice graded ring, i V € V(R)t a n d let xoy = yox = 0. It is easy to give an example showing that supp c (a;) and supp c (y) may happen to be distinct. However, we show that subc(a^) = subc{y)x

Lemma 5.2 If C is a semilattice, R = ®cecRc o- semilattice graded ring, x,y £ V(R) and xoy = yox = 0, then subc(x) = subc(y)73

74

Groups of Units

Proof. Suppose to the contrary that there exists an element m belonging to only one of the sets subcOc) and subc(y)- Choose a maximal element m with this property. We may assume that m £ subc(y)- Put A = sut>c(x)nm and B = subc(y)nm. By the choice of m, we get B = {m}UA. Denote by p the product of all elements of A. Since subc(^) is a subsemigroup of C, it is clear that A is also a subsemigroup of C, and so p £ A. Therefore x~ = x~ and y— = ym + y~. By Lemma 5.1, we get x c + V x~y^ = x^ + y~— x^y^ = 0, whence ym + x^ym = 0. Since m

a

m

m^m

p

a

p

p P

m^'

l

x £ V(R), Lemma 5.1 shows that x~ £ V(R), and so 1 + x~ £ [/(.R1). Therefore ym = 0, a contradiction. • Every band B is a semilattice C of rectangular bands Hc, where c £ C. For b £ B, denote by b the image of b in C. Let R = (BbeBRb be a band graded ring, r = ^2beB rb £ R. If we put Rc = X^= c Rt>, t n e n R — ®cecRc is a semilattice graded ring, and r c = Yjb=crb- ^ e n e e d t n e following Lemma 2.3 of [Munn (1992)]. Lemma 5.3 If R — (BbeBRb is a special band-graded ring, b £ B and x, y £ R^, then xlby = xy, Now let R = (BbeBRb be a special band-graded ring. For a,b £ B with a > b, define fg:Ra—> Rb by fb(x) = hxlb (where x £ Ra). The following basic properties of these mappings were established in [Munn (1992)], Lemma 2.2. Lemma 5.4 Let R = (BbeBRb be a special band-graded ring, and let a,b £ B with a >b. Then (i) fjj is a homomorphism; (") fbfa =_fb f°r anyceB with c > a; (hi) if a = T> then fjj is an isomorphism, with inverse

f\.

Lemma 5.5 Let R = (BbeBRb be a special band-graded ring, b £ B, and let f : RT- —> Rb be defined by f(x) = lbxlb for x £ R~. Then f is a homomorphism of R?- onto RbProof. The equality f(x + y) = f(x) + f(y) is obvious. Hence it suffices to check f(xy) = f(x)f(y) for x £ Rg,y £ Rh, where g,h £ b. Let B be a semilattice C of rectangular bands Hc, and let b £ Hc for c £ C. Put

75

H = Hc. Then lbX,ylb £ RH, and so Lemma 5.3 yields f(xy) = lbxylb = lbxlbylb = f(x)f(y), completing the proof. • Lemma 5.6 Let R = (BbeBRb be a special band-graded ring, and let B be a semilattice C of rectangular bands Hc. For each c £ C choose an element e(c) in Hc and define a mapping f : R —> r i c e c -^e(c) by the rule f(x) = ^2C£c le(c)a;'7~\le(c) for x £ R. Then f is a homomorphism of R into the direct product D = Ilcec ^e(c)> and the kernel of f is a nil ideal of R. Besides, if C is finite, then f(R) = D. Proof. Lemmas 5.1 and 5.5 show that / is a homomorphism. Now take any x £ Ker(/). We claim that x is nilpotent. Let m be a maximal element in subc(a;). Look at any a,b £ supp B (a;). If a ^ m or b ^ m, then agb ^ m for any g £ supp B (x). Further, if a = b = m, then Lemma 5.3 implies xaxmxb = xale{m)xml^m)Xb = a; a l e ( m ) a;---l e ( m )a; 6 = 0. Therefore (x3)m = 0, and so subc(a; 3 ) C sub(x)\{m}. Hence the induction on |subc(x)| shows that x is nilpotent. Now assume that C is finite and take any c € C. Put z

= l[0-e(c) ~ le(d)), d R e tC

(6.1)

Re G K for all e G E{S) -£=> R G K

(6.2)

and equivalence

have been considered. The main theorem of this section shows that, for ring classes K. with certain natural closure properties, as soon as the relation (6.1) or (6.2) has been verified for rings graded by finite groups, it immediately holds for rings graded by finite groupoids (Theorem 6.1). For rings graded by finite groups results of this sort are deducible with the use of duality theory of Cohen and Montgomery [Cohen and Montgomery (1984)]. A class K. of rings is said to be closed under finite sums of one-sided ideals if and only if, for every ring with right (or left) ideals A,B G /C, it follows that A + B G tC. Our proof applies to both (6.1) and (6.2), and so we combine "if and only if" and "provided that" parts in one theorem. Theorem 6.1 ([Kelarev (1995c)]) LetK be a class of rings which contains all rings with zero multiplication and is closed for homomorphic images, 77

78

Finiteness

right and left ideals, ring extensions. equivalent:

Conditions

Then the following assertions are

(i) for each finite groupoid S, an S-graded ring R — ®s£sRs belongs to K. provided that (if and only if) Re belongs to K for every idempotents e of S; (ii) for each finite semigroup S, an S-graded ring R = ®sesRs belongs to K. provided that (if and only if) Re belongs to K, for every idempotent e of S; If, moreover, K, is also closed for finite sums of one-sided ideals, then the following is equivalent to the above assertions: (iii) for every finite group G with identity e, a G-graded ring R = ®geGRg is in K provided that (if and only if) Re G K. Proof. The implication (i)=>(ii) is trivial. Assume that (ii) holds. We claim that then K is closed for finite sums of one-sided ideals. Clearly, it suffices to consider a ring A which is the sum of its two right ideals M and N from K. Let L = {a, b} be the two element semigroup such that ab = a = a2, ba = b = b2. In the semigroup ring A[L] consider subrings Ra — Ma and Rb = Nb. Then R = Ra + Rb is L-graded. Since Ra = M and Rb = N belong to AC, it follows from (ii) that R G K,. It is easily seen that I = {k(a — b) | k G M n N} is an ideal of R and R/I = A. Given that K is closed for homomorphic images, we get A G K, as required. Since (ii)=4>(iii) is also easy, it remains to prove (iii)=>(i) with the extra hypothesis that K. is closed for finite sums of one-sided ideals. Let S be any finite groupoid, R = ®szsRs a n -S-graded ring and / a homogeneous two-sided ideal of JR. Given that K, is closed for ideals and homomorphic images, it is easy to prove that R is a counterexample for (i) if, and only if, either / or R/I is a counterexample for (i). Suppose now that R = ®s&sRs is a counterexample for (i) with \S\ minimal. We claim that, given any s G 5 and any additive subgroup A of Rs such that supp(yl.R) ^ S, there is a two-sided homogeneous ideal I of R such that A C 7, I G K, and Ie G K. for every idempotent e € E(S). Obviously, I is not a counterexample for (i). As a consequence, we shall be able to factor out I, and get a new counterexample to (i) with A = 0. If AR = 0, we take I = RlA. Then I2 = 0, and so I G K and It G /C, for every t G S.

Groupoid-Graded

Rings

79

If AR = P ^ 0 then, by the minimality of \S\, P cannot be a counterexample to (i). In the "provided that" part of our theorem, R being a counterexample implies that Re is in /C, for every e £ E(S). Then Pe £ K. for every e £ E(S), because Pe is a right ideal in Re. Since P satisfies (i), we get P £ fC. (In the "if and only if" version of the theorem, R being a counterexample implies that either R £ K or Re is in fC, for every e £ E(S). If R £ JC, then P £ K. because P is a right ideal of R. If i? e is in /C, for every e e JS(S'), then Pe is also in /C, and so P £ /C, again.) But supp(i? x P) ^ S1, for every x £ S, and the same argument applied to the additive subgroup RXA contained in Rxs tells us that RXA and (RxA)e are in K, for every idempotent e £ S. Given that fC is closed for finite sums of two-sided ideals, it follows that I = RlP = P + Y^xes RxP is a homogeneous two-sided ideal with the desired properties. If now s £ S and sS ^ S, then supp(RsR) C sS =£ S and, putting A = Rs above, we get a counterexample for (i) whose s-th homogeneous component is zero and hence can be graded by the set T = S\{s}. We can introduce a multiplication on this set, make it a groupoid and get a contradiction to the minimality of \S\. Therefore sS = S and, by changing sides in this argument, we get Ss = S. Thus S is a left and right simple groupoid. We claim that it is also associative and it is thereby a semigroup. Indeed, if (st)x / s(tx) then RsRtRx Q R(st)x^Ps(tx) implies that RsRtRx = 0 and so s\ipp(RsRtR) ^ S, because S is finite. By applying the above paragraph with A = RsRt, we may assume RsRt = 0. Again using the same reduction with A = Rs, we can also assume Rs — 0, because supp(RsR) ^ S. This yields a contradiction with the minimality of | 5 | . So our claim has been established. By Lemma 1.1, 5 is a group, and we get a contradiction with (iii), which completes the proof. • The following example shows that the closedness restrictions on K are essential in Theorem 6.1. Example 6.1 The class A4 of Brown-McCoy radical rings satisfies implication (6.1) in (iii), but not in (ii). It is well-known that M. contains all rings with zero multiplication and is closed for ring extensions, homomorphic images and ideals ([McCoy (1964)], § 37). Theorem 5 of [Grzeszczuk (1985)] says that, for every finite group G with identity e, each G-graded ring R belongs to M provided that Re £ Ai. Take a simple non-Artinian domain R with unity (for example, the Weyl

80

Finiteness

Conditions

algebra Ai). Pick two different maximal right ideals M and N in R. It is proved in [Beidar (1982)], Lemma 2, that M and N are simple. Since R has no nonzero idempotents, M and N are rings without identities, and so they are Brown-McCoy radical rings. However, R = M + N is BrownMcCoy semisimple. Therefore M is not closed for sums of two right ideals. It follows from the second paragraph of the proof of Theorem 6.1 that M. does not satisfy (ii). For groupoid gradings the terms "graded ring with finite support" and "ring graded by a finite groupoid" are the same. As a consequence, conditions (i), (ii) and (iii) of Theorem 6.1 are equivalent, under the given hypotheses, to the corresponding statements obtained by replacing "ring graded by a finite groupoid (semigroup, group)" by "groupoid (semigroup, group)-graded ring with finite support," and so the theorem can be rewritten for graded rings with finite supports. In the "if and only if" version we can slightly weaken the closedness restrictions imposed on the class in the hypothesis of Theorem 6.1. Indeed, the example in the last paragraph of the proof of necessity of [Kelarev (1991a)], Theorem 6.1, shows that every class K, which contains all rings with zero multiplication, is closed for ring extensions and homomorphic images and satisfies the "if and only if" version of (ii) is also closed for one-sided ideals. For completeness we include a proof of this fact. Consider a ring A G K with a right ideal / . Let L = {c, d} be a semigroup such that cd = c = c 2 , dc = d = d?. Then R = Ac + Id is L-graded. It is readily verified that N = {i(c — d) \ i G 1} is an ideal of R and iV2 = 0. Since ./V G K and R/N =* A G /C, we get R G K. The "if and only if" version of (ii) yields I = Rd G /C, as required. Thus, in the "if and only if" case of Theorem 6.1 the closedness of K. for right and left ideals can be moved from the hypothesis of the theorem to the extra restrictions before (iii). Suppose that R is an 5-graded ring and Rs is a subring of R. If s 0 E(S), then fi2 C Rs n Rs2 = 0. Since every class of Theorem 6.1 contains all rings with zero multiplication, this theorem and several corollaries can be rewritten by replacing "every e in E(S)" by "every Rs which is a ring". For the class of all quasiregular (locally nilpotent, Baer radical) rings the properties in the hypothesis of Theorem 6.1 are well-known. In particular, it is known that Jacobson (Levitzki, Baer) radical of a ring contains all quasiregular (locally nilpotent, Baer radical) left and right ideals of the ring, and therefore these classes are closed for sums of one-sided ideals. For

Groupoid-Graded

Rings

81

Pi-rings all the necessary closedness properties are obvious except the very difficult fact that every sum of two right (or left) ideals satisfying polynomial identities is a Pi-ring. This was proved by Rowen [Rowen (1976)]. Jaegermann and Sands established that the class of Jacobson rings is an iV-radical class, and that all such classes are closed for left and right ideals (see [Jaegermann and Sands (1978)], p. 348 and Theorem 11). For semilocal, right or left perfect, semiprimary rings all properties were proved by Clase and Jespers [Clase and Jespers (1993)]. For the classes of semilocal, right or left perfect, semiprimary, nilpotent, locally nilpotent, T-nilpotent, Baer radical, quasiregular and P.I. rings the "if and only if" versions of (iii) are also known: see [Jensen and Jondrup (1991)] for right or left perfect and semiprimary rings; [Beattie and Jespers (1991)] for right or left perfect, semilocal, and semiprimary rings; [Cohen and Rowen (1983)] for nilpotent rings; [Cohen and Montgomery (1984)] for quasiregular, Baer radical, and locally nilpotent rings; [Kelarev (1993e)] for Pi-rings. Note that for nilpotent and T-nilpotent rings assertion (iii) (and even Corollary 6.1 below) follows from Ramsey theorem (see [Kelarev (1994a)]). Combinatorial proofs are also possible for the classes of locally nilpotent and Baer radical rings ([Kelarev (1994a)]). "Provided that" version of (iii) for Jacobson rings is contained in [Passman (1989)], see also [Clase and Jespers (1993)]. Corollary 6.1 Let K, be the class of all semilocal (right perfect, left perfect, semiprimary, nilpotent, locally nilpotent, T-nilpotent, Baer radical, quasiregular) rings, S a finite groupoid, R — ®s^sRs an S-graded ring. Then R 6 K if and only if all Re belong to K, for all e £ E(S). Besides, R is a Jacobson ring provided that Re is a Jacobson ring for every e £ E(S). Corollary 6.2 Let K be a class of rings which contains all rings with zero multiplication and is closed under subrings, homomorphic images, ideal extensions and finite sums of one-sided ideals. Then, for any finite groupoid S, each S-graded ring R = ®s^sRs belongs to K. if and only if Re is in K. for every e in E(S). Proof. The result is true for S a group by [Cohen and Montgomery (1984)], Theorem 3.5, as was noted in [Kelarev (1993e)], Lemma 4. Hence the corollary follows from Theorem 6.1. • Corollary 6.3 ([Kelarev (1995c)]) Let AC be the class of all semilocal (right perfect, left perfect, semiprimary, nilpotent, locally nilpotent, T-nilpotent,

82

Finiteness

Conditions

Baer radical, quasiregular) rings, S a semigroup, R = ®s^s^s an S-graded ring with finite support. Then R 6 K. if and only if Re £ fC for every e in E(S). Corollary 6.4 Let S be a semigroup, R = (BsesRs o,n S-graded ring. If Re is semilocal for every e in E(S) and if Rs C J{R) for all but finitely many s in S, then R is semilocal, too. Let us include only one corollary concerning rings with not necessarily finite supports. Corollary 6.5 Let S be a periodic semigroup with a finite number of idempotents and only finite subgroups, and such that all nil factors of S are nilpotent. Let R = (BsesRs be an S-graded ring. Then R is semiprimary if and only if Re is semiprimary for every e in E(S). Proof. By Theorem 1.9, S has a finite ideal chain with finite or nilpotent factors. If / is an ideal of S, then R is an extension of Rj by R/Ri, and R/Ri is graded by S/I. Hence it follows by induction on the length of the ideal chain of S that it suffices to prove the corollary for all factors of S. For finite factors Theorem 6.1 gives the result. For nilpotent factors the claim is obvious. • Similar corollaries can be written for other properties mentioned in Corollary 6.1. Known examples of semigroup rings show that all restrictions on semigroup S in Corollary 6.5 are essential. 6.2

Structural Approach of Jespers and Okninski

Let us begin this section with two best results on finiteness conditions achieved so far. Theorem 6.2 ([Jespers and Okninski (1995)]) Let G be a group, and let R be a G-graded ring with J{Re) Q Jgr{R) ^ R- Then the G-graded ring R/Jgr{R) is semilocal (respectively, left perfect, semiprimary, left Artinian) if and only if the following conditions are satisfied: (i) Re/J(Re) = M n i ( D ( 1 ) ) x •••Mnr(D(r)), where every D(i) is a division ring. (ii) For any complete set of orthogonal idempotents eu of Re/J(Re), where \ 2 and the result holds for smaller nilpotency indices. Consider the ring R = R/I2 with ideal 7 = 7/7 2 . R and 7 satisfy the hypotheses of the lemma and I2 = 0. Hence, by the n = 2 case, supp(T^) is finite. This implies that there are only finitely many s £ S such that Rs CI2. Since 7 2 has nilpotency index less than n, we conclude by the induction hypothesis that supp(i?) is finite, as required. • Lemma 6.5 Let S be a semigroup and let R be a right Artinian S-graded ring. Let s be a non-periodic element of S. Then Rs consists of nilpotent elements.

86

Finiteness

Conditions

Proof. Let Se be the semigroup with an identity e adjoined. Denote by R1 the ring obtained by adjoining an identity 1 to R in the usual way. If we put Re = Z, the subring generated by 1, then R1 = (BteS'Rt is an 5 e -graded ring. Suppose to the contrary that there exists an r £ Rs which is not a nilpotent element. For any non-negative integer m, denote by Im the right ideal of R1 generated by 1 — r2 . Then Im D Im+i because 1 — r 2 = (1 — r 2 )(1 + r 2 ). Given that R is right Artinian, there is some k > 0 such that IknR = Ik+i tlR = .... However, R1 = Ik + R = 7fc+1 +R= .... It follows that Ik = Ik+i = Since r is not nilpotent, neither is r 2 . Replacing s2 by s and r2 by r we may assume that k = 0 and that IQ = I\ = .... Since 1 — r2 £ I\ = IQ, there exists an element w £ R1 such that 1 — r = (1 — r2)w. Denote by B the subsemigroup generated in Se by e and s, and let C = S \ B. Note that R1 = RB © Re, s o for any x € R1, we can write x = XB +%C uniquely with XB € RB and xc G Re- Consider the equation 1 — r = (1 —r2)(wB+wc). Since 1, r, and % are elements of the subring RB, if we take components in Re, we see that wc = (r 2 toc)c^ But r2 is a homogeneous element of R, so | supp(r 2 wc)c| < | suppr 2 ?!^! < |suppwc|- This and the previous equality imply that wc = r2wc, and so 1 — r = (1 — r2)wBNow write WB = WQ + wx + W2 + • • • where wo £ Re and wn £ Rsn for n > 0. Comparing the homogeneous summands in the equation 1 - r = (1 - r2)(w0 + wi + w2 + ...) we see that WQ = 1 and w„ = ± r n for n > 0. Since w„ = 0 for n sufficiently large, this contradicts the assumption that r was not nilpotent. • Proposition 6.2 ([Clase et al. (1995)]) Let S be a semigroup with no infinite subgroups and let R be a right Artinian S-graded ring. Then there exist finitely many elements xi, x^, • • •, xn £ S such that R = ^T(R) +RXl + H-X2

I • • • 1 Kxn

•

Proof. Since R/J{R) is semisimple Artinian, there are ideals W\, W%, . •., Wi of R, containing J(R) such that each Wi/J(R) is simple Artinian and R/J(R) = ®\=1Wi/J(R). It suffices to show that each W* C J(R) + Rx for some finite set X C S. Fix W = Wi, and let W = W/J(R). We will denote the image of an element r £ R (or a subset Y C R) under the composite of the natural

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maps R —> R/J(R) -+ W by r (or y ) . Since VK is simple Artinian, it is isomorphic to a ring of p x p matrices over some division ring. Notice that R = W. Consider the right ideals of R of the form RxSi for x G S. Since R is right Artinian, we can choose x such that Rxs^ is minimal subject to the condition Rxsi ^ 0. We claim that there is a u G xS1 and an a G Ru such that a is not nilpotent. For otherwise, the semigroup H of homogeneous elements of Rxsl is such that its image H in W is nil. By Theorem 1.12, Rxs^ is nilpotent, each element being a sum of elements of H. But this is not possible, since RxSi is a nonzero right ideal of the simple Artinian ring W. By Lemma 6.5, u must be a periodic element of S. So some power of u is an idempotent e. Note that Re / 0 since it contains a power of a. But eS1 C xS1. So by choice of x, we must have Resi = RxS1- Replacing a by the appropriate power, we may assume that a G Re and a is not nilpotent. Regarding a £ W as a, matrix, there must be a q such that aq and a2q have the same rank. Replacing a by a 9 , we still have a G Re and we may further assume that o and a2 have the same rank as matrices. This means that there is an idempotent / in W such that faf = a and / has the same rank as a (that is, a lies in the maximal subgroup of W containing / ) by Theorem 1.13. Hence, aRa = fRf = fWf is a corner of the matrix ring

W. We claim that it is sufficient to find a finite set X C S such that fWf C Rx- For we may choose a matrix basis e^ for W in such a way that / = e n + e^2 + • • • + e-u (where i is the rank of / ) , and then W = Y^lj=i enfWfeij. Ifr,j is an element of R such that fij = e^ for each i and j , then eufWfeij C RY(i,j) where Y(i,j) = supp(rji)Xsupp(rij) is a finite set. Hence, W C RY where Y = UI\'=i ^(*> j ) is finite, and it follows that W C J(R) + RY as desired. So we must find such a finite set X. In fact, we will take for X the subgroup G = W(]5]) of units of the monoid eSe. By the hypothesis on S, such a G must be finite. Let z G S and let b G Rz be a homogeneous element such that aba ^ 0. We claim first that there is a y G S and a homogeneous element c G Ry such that abaaca is not nilpotent. For otherwise, as before, abaaRa is a nilpotent right ideal of aRa, and therefore abaaRa = 0. Since aRa is simple Artinian, this implies aba = 0, a contradiction.

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Now, ezeye = ezeeye = supp(abaaca) is periodic by Lemma 6.5 since abaaca is not nilpotent. Hence, there is a v > 1 such that e' = (ezeye)v is idempotent. Note that e' = ee' = e'e. We have e'S1 C e-S1. But 0 ^ (abaaca)v € i?e'i s o by the minimality of Res^, we must have i?e<si = i? e gi. Because 0 ^ o € i? e , we must have eS'1 = e'5 1 , and it follows that e = e'e = e'. Hence eze has a right inverse in the monoid eSe. Similarly, eze has a left inverse in eSe, and so eze € G. We have shown that for all homogeneous b G R, if a6a ^ 0, then aba € i?G. Hence, / W / = ai?a C RQ. By the assertions above, this completes the proof. D Now we can prove Theorem 6.4. Proof. By [Murase and Tominaga (1978)], Theorem 8, the ring R2 is both right Artinian and right Noetherian. Since R/R? is a nilpotent right Artinian .S-graded ring, it follows from Lemma 6.4, that supp(i?/i? 2 ) is finite. So it suffices to prove that R2 has finite support. Replacing R by R2, we may assume that R is right Artinian and right Noetherian. By Proposition 6.2, there is a finite subset XQ C S such that R = J{R) + Rx0- Since R is right Artinian, J{R) is nilpotent; let m be its index of nilpotency. Fix k, with 1 < k < m. Since R is right Noetherian, the quotient J(R)k/J(R)k+1 is finitely generated as a right i?-module, so there are elements b\, 62, • • •, bi G J{R)k such that l

J(R)k = J2 (btZ + hR) + J(R)k+1. i=\

Note that btR = biJ(R) + hRXo. Since biJ(R) C J(R)k+1 and since k+1 the support of each bi is finite, we see that hR C J(R) + RXk. where Xk,i = supp(bi)X0 is finite. Therefore, J(R)k C RXk + J(R)k+1 where Xk = Ui=i (Xk,i U supp(fcj)) is finite. Combining these sets for 1 < k < m, we see that J{R) C RXl + RX2 + ... i?x m _i) a n < i therefore, the support of R is contained in the finite set X0l)XlU...UXm-i. • An examination of the proof shows that we only require that each subgroup G of S intersects the support of R finitely. Since bands have only trivial subgroups, we get

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Corollary 6.8 ([Kelarev (1995a)]) Let B be a band and let R be a right Artinian B-graded ring. Then the support of R is finite.

6.3

Finiteness Conditions and Homogeneous Components

Let us begin with the following generalization of a well-known theorem by Chin and Quinn. Theorem 6.5 ([Dascalescu and Kelarev (1999)]) Let S be a semigroup which is a disjoint union of polycyclic-by-finite groups, satisfies d.c.c. for principal ideals, and does not contain nontrivial left zero bands. Let R = 0 s e S Rs be an S-graded ring, and let M = © s £ 5 Ms be an S-graded R-module. Then M is graded Noetherian if and only if it is Noetherian. The remaining main theorems of this section show that, for several interesting ring properties, if a ring R is graded by an epigroup S with certain restrictions (like the finiteness of the number of idempotents, etc.), then the property of R is determined by the group-graded subrings RQ or homogeneous components Re. All these theorems are complete in the sense that all the restrictions on the underlying semigroup S are essential. The following technical concept plays key roles in these results. Let K. be a class of rings, S a semigroup. We say that K is S-closed if /C contains each 5-graded ring R — ®sSsRs, provided that all subrings Re belong to K, for all idempotents e € S. Also, we include analogous results relating the properties of an S-graded ring R to the properties of subrings RQ of R graded by the maximal subgroups G of S. The theorems of this section have two equivalent conditions (i) and (ii) for a semigroup S. Condition (i) is always a ring property satisfied in 5-graded rings, and condition (ii) gives a clear description of a certain type of semigroups. Prom the point of view of graded ring theory, implication (ii)=S>(i) is more interesting. It shows that whenever a ring can be graded by a semigroup with (ii), a certain ring property of the graded ring can be determined as in (i). On the other hand, implication (i)=^(ii) shows that the class of semigroups included in (ii) cannot be extended, and for more general gradings (i) is not applicable. Theorem 6.6 ([Kelarev (1995d)]) For a semigroup S, the following conditions are equivalent:

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(i) Every S-graded ring R = © s € gi? s is right Artinian (Noetherian), provided that all group-graded subrings RQ are right Artinian (respectively, Noetherian) for all maximal subgroups G of S; (ii) S is the union of a finite number of groups and S does not contain the two-element right zero band. A few lemmas are required for the proof. Lemma 6.6 / / S is not a union of groups, and if a class K contains 0 and every S-graded ring R such that all RQ belong to K, for all maximal subgroups G of S, then K, contains all rings with zero multiplication. Proof. Take any ring R with zero multiplication. Pick an element t in S which does not belong to any subgroup of S. If we put Rt = R, and Rs = 0 for all s £ S, then R becomes an S-graded ring. Since RQ = 0 € /C for all subgroups G of S, it follows that Re K.. • Lemma 6.7 Let S be a semigroup such that the class K. of all semilocal (semiprimary, right or left perfect, right or left Artinian, right or left Noetherian, nilpotent, PI) rings (or the class of all rings with nilpotent Jacobson radicals) contains each S-graded ring R, provided that RQ G K. for every subgroup G of S. Then the number of idempotents of S is finite. Proof. Suppose to the contrary that S has pairwise distinct idempotents ei, e-i, It is known that every class K, in the hypothesis contains rings Pi,P2,... such that the direct sum © ^ . P * does not belong to K. For n = 1,2,..., let R€n = Pn. For all the other s £ S put Ra = 0. Then R = ®sesRs is an S-graded ring. Fix a subgroup G of S. Clearly G can contain at most one of idempotents e i , e 2 , . . . . Hence either RG = 0 or RQ = Re„ for some n. Therefore RQ 6 /C. However, R £ K,. This contradiction completes the proof. • Lemma 6.8 If L is a right (left) zero band, and K is an L-closed and closed for homomorphic images class of rings, then K, is closed under finite sums of left (right) ideals. Proof. Take any ring R which is the sum of a finite number of left ideals Ra, s £ S, belonging to fC. Introduce an operation on S such that S becomes a right zero band. Consider the semigroup ring RS. Clearly, the subring T — ®3&sRss is S-graded with components Rss isomorphic to Rs.

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Given that K is 5-closed, we get T G K. Therefore R € K., because R is a homomorphic image of T. D Lemma 6.9 Let IC be a class of rings closed under ideal extensions, and let S be a semigroup such that S° has a finite ideal chain 0 = S0 C Si C S2 C • • • C Sn = S°. Then K. contains all S-graded rings with subrings of the form RQ in IC, where G runs over all maximal subgroups of S, if and only if the same can be said of every factor of the chain. Proof. The 'if part. Suppose that, for all i = 1 , . . . , n, the class K contains all 5j/5j_i-graded rings with subrings of the form RQ in K. Consider an S-graded ring R = ®s^sRs s u c h that RQ G fC for all maximal subgroups G of S. For 1 < i < n, the quotient ring RsJRsi^! is an S^/S^i-graded ring with homogeneous components isomorphic to the components of R. For every maximal subgroup G of S, either G C Si-i and Ro/Rsi-i — 0; or G n Si-i = 0 and Rc/Rsi-! — RG € IC. Hence our assumption implies that Rsi/Rsi-i € IC- Since the class IC is closed for ideal extensions, easy induction on n shows that R G IC. The 'only if part. Consider an Si/St-j-graded ring R with homogeneous component Ro. It is an ideal extension of RQ by a contracted S'j/5'j_1graded ring. It remains to note that every contracted S , j/S , i _i-graded ring is S-graded. • Now we can prove Theorem 6.6. Proof. The implication (i)=>(ii) is easy. By Lemma 6.6, S is a union of groups. Lemma 6.7 tells us that S has only a finite number of idempotents, and so it is a union of a finite number of groups. Since A is not closed under finite sums of left ideals, Lemma 6.8 shows that S does not contain the two-element right zero band. Conversely, if S is the union of a finite number of groups and S does not contain the two-element right zero band, then S has a finite ideal chain whose factors are unions of their right ideals which are groups with the same zero adjoined. Given that RG & A for all such subgroups, we get R € A by Lemmas 1.5 and 6.9. • Corollary 6.9 ([Kelarev (1995d)]) For a semigroup S, the following conditions are equivalent:

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(i) Every S-graded ring R = (BsesRs is right Artinian provided that all components Re are right Artinian for all idempotents e of E(S); (ii) S is a finite left regular band.

(Noetherian), (Noetherian)

Lemma 6.10 If S is not a band, and K, is an S-closed class of rings containing 0, then K. contains all rings with zero multiplication. Proof. Take any ring R with zero multiplication. Pick t e S such that t T^ t2. If we put Rt = R, and Rs = 0 for all s £ S, then R becomes an S-graded ring. Since Re = 0 € K. for all e e E(S), it follows that R e K,. • Now we prove Corollary 6.9. Proof. Assume that (i) holds. Lemma 6.10 (i) shows that S is a band. By Lemma 6.7 S is finite. Since A is not closed under finite sums of left ideals, it follows that S does not contain the two-element right zero band. By Theorem 1.16 S is a left regular band. Conversely, let 5 b e a finite left regular band. Then all subgroups of S are singletons, and Theorem 6.6 completes the proof. • Corollary 6.10 ([Kelarev (1994b)]) Let B be a semilattice, R = (BbeBRb a special band graded ring. Then R is right Artinian (right Noetherian) if and only if B is finite and all the R/, are right Artinian (right Noetherian). Corollary 6.11 ([Kelarev (1994b)]) Let R = (BbeBRb be a special bandgraded ring. If R is right Artinian (right Noetherian), then B is finite and all the Rf, are right Artinian (right Noetherian). Theorem 6.7 ([Kelarev (1995d)]) For a semigroup S, the following conditions are equivalent: (i) each S-graded ring R is semilocal (semiprimary, right perfect, left perfect) if and only if all group-graded subrings RQ are semilocal (semiprimary, right perfect, left perfect) for all maximal subgroups

GofS; (ii) S is an epigroup with a finite number of idempotents and every nil factor of S is locally nilpotent (nilpotent, right T-nilpotent, left T-nilpotent). Lemma 6.11 ([Kelarev (1989c)], Lemma 4) Let S be a semigroup which is not locally finite. Then there exists an S-graded ring R such that R\ = 0 for every e € E(S), and R is not quasiregular.

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Lemma 6.12 Let S be a semigroup such that the class of all semilocal (semiprimary, right perfect, left perfect, nilpotent, right T-nilpotent, left T-nilpotent, Jacobson, quasiregular, Baer radical) rings (or the class of all rings with nilpotent Jacobson radicals) contains each S-graded ring R, provided that RQ = 0 for every subgroup G of S. Then S is an epigroup. Proof. Suppose to the contrary that S is not an epigroup. It means that S has an element t generating a subsemigroup T = {t,t2,...} which does not intersect any subgroup of S. Consider the ring R of polynomials over C (in the case of the class of all rings with nilpotent Jacobson radicals, we consider polynomials over any locally nilpotent but not nilpotent ring) with non-commuting variables x\,X2,... without free terms. For n = 1,2,..., denote by 7?t" the set of all homogeneous polynomials of degree n. Put R3 = 0 for s G S\T. Then R = ®sesRs is S-graded. Clearly, RQ = 0 for every subgroup G of S, and so R belongs to the class in the hypothesis of this lemma. However, it is well-known that all the classes included in lemma do not contain R. (Note that R is not Jacobson. For example, it can be homomorphically mapped onto the Golod ring, that is the first example of a nil ring which is not locally nilpotent.) • Lemma 6.13 Let S be a semigroup such that the class of all semilocal rings contains each S-graded ring R, provided that RQ = 0 for every subgroup G of S. Then every nil factor of S is locally nilpotent. Proof. Suppose that a nil factor F of S is not locally nilpotent. There exist ideals I C J of S such that F = J/I. Consider the contracted semigroup ring R = CQF. By Lemma 3.1, R is not semilocal. Put Rs = Cs for s G J\I, and Rs = 0 for s G JU (S\J). Then R = ®s€SRs is S-graded and every Re = 0 is semilocal for each subgroup G of S. This contradiction completes the proof. • Lemma 6.14 Let S be a semigroup such that the class of all semiprimary (or nilpotent) rings (or the class of all rings with nilpotent radicals) contains each S-graded ring R, provided that RQ = 0 for every subgroup G of S. Then every nil factor of S is nilpotent. Proof. Suppose to the contrary that a nil factor F of S is not nilpotent. There exist ideals / C J of S such that F = J/I. Consider the contracted semigroup ring R = C0F. Put Rs = Cs for s € J\I, and Rs = 0 for s G IU(S\J). Then R = ®seSRs is S-graded. Since J/I is nil, clearly J\I

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has no idempotents, and so Re = 0 for every e e E(S). The choice of 5 forces R to belong to the class of rings in the hypothesis. Since F is not nilpotent, evidently, R is not nilpotent, either. Lemma 3.1 shows that R is not semiprimary. Further, take any locally nilpotent ring L which is not nilpotent. Consider the contracted semigroup ring R = LQF. Put Rs = Ls for s € J\I, and Rs = 0 for s e / U (S\J). Then R = ®sesRs is S-graded. Since Re = 0 for every e £ E(S), the radical ^(.R) is nilpotent. Obviously, R is locally nilpotent, and so J{R) = R. However, R is not nilpotent. This contradiction completes the proof. • Lemma 6.15 Let S be a semigroup such that the class of all right (left) T-nilpotent or right (left) perfect rings contains each S-graded ring R, provided that RQ = 0 for every subgroup G of S. Then every nil factor of S is right (left) T-nilpotent. Proof. Consider any nil factor F of S and introduce the same S-graded ring R as in the proof of Lemma 6.13. If F is not right (left) T-nilpotent, then clearly R is not right (left) T-nilpotent, and Lemma 3.1 implies that R is not right (left) perfect. • Lemma 6.16 Let S be a semigroup with a sequence of elements e\, e 2 , . . . such that, for any 1 < k < I < m, the products e^ . . . e; and e ; + i . . . em are not contained in the same subgroup of S. Then there exists an S-graded ring R such that RG is a ring with zero multiplication for every subgroup G of S, but R is neither right nor left T-nilpotent. Proof. We shall construct an 5-graded ring which is not left T-nilpotent. (The case of right T-nilpotency is dual.) Let F be the free semigroup with the set X = {#1, £2, • • •} of generators. Let I be the set of all products Xix... Xin such that e j , , . . . , ej n is not a subword of the infinite word ei,e2, Obviously, I is an ideal of S. Consider the contracted semigroup ring R = Co(F/I). Put R3 = Cs for s € F\I, and Rs = 0 for s e L Then R = (BszsRs is S-graded. For all n, x\... xn $ I, and so R is not left T-nilpotent. Fix a subgroup G of S. For any elements a, b 6 RQ of the form a = Xix ... Xim and b = x^x ... Xjn, where all Xik, Xjl £ X, we claim that ab = 0. Indeed, if ab ^ 0, then x^ .. .XimXj1 .. .Xjn does not belong to I, and so e , ! , . . . , ej m , ejx,..., ejn is a subword of the word ei, e2,.... Given that

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a,b £ RQ, we get e^ . . . ej m £ G, and ejl... e Jn G G, which contradicts the choice of the sequence e\, e 2 , Thus .RQ = 0 for every subgroup G of the semigroup S. • Lemma 6.17 Let S be a semigroup containing an infinite idempotent chain. Then there exists an S-graded ring R such that RQ is a ring with zero multiplication for every subgroup G of S, but R is neither right nor left T-nilpotent. Proof. Every infinite idempotent chain contains an infinite descending or ascending subchain. If S contains idempotents e\ > e% > ... or e\ < e-2 < ..., then it is routine to verify that both these sequences satisfy the requirements of Lemma 6.16, which completes the proof. • Lemma 6.18 Let S be a semigroup such that every S-graded ring R is left (right) T-nilpotent, provided that RG = 0 for every subgroup G of S. Then S satisfies the descending chain condition for principal right ideals (respectively, the descending chain condition for principal left ideals). Proof. Suppose to the contrary that S contains an infinite descending chain of principal right ideals siS1 D S2S1 D ..., for some si,S2, • • • £ S. Then, for n = 1,2,..., there exists en G S such that snen = s„+i. It easily follows that the sequence ei, ei,... satisfies the requirements of Lemma 6.16, which completes the proof. • Lemma 6.19 Suppose that the class of right or left T-nilpotent rings contains each S-graded ring R, provided that RG = 0 for every subgroup G of S. Then every completely 0-simple factor of S° has a finite sandwich matrix. Proof. Take any completely 0-simple factor F of 5°. There exist ideals / C J of S such that F = J/I. Represent F as a Rees matrix semigroup M°(G; I, A; P). Suppose to the contrary that P is infinite. Then / U A is infinite. We consider only the case where |7| > 00, since the case |A| > 00 is analogous. Choose pairwise distinct elements ii,i2,... in J. Since every column of P has a nonzero entry, for n = 1,2,..., there exists A„ £ A such that P\nin+i ¥" 0- F i x a n Y 9 € G and introduce elements e„ = (g,in,^n) for n = 1,2,.... For i G I, put Tt = U A e A (G,i,A). If 1 < m < n, then G Tim. Clearly, all T,TO are pairwise disjoint, and each nonzero

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subgroup of S is contained in one Tj m . Therefore the sequence e\, e2, • • • satisfies all requirements of Lemma 6.16, which completes the proof. • Lemma 6.20 Let S be a semigroup such that the class of all Baer radical (Jacobson) rings contains each S-graded ring R, provided that all groupgraded subrings Ra are Baer radical (Jacobson) for all maximal subgroups G of S. Then S has no infinite descending idempotent chains. Proof. We shall record the complete proof for Jacobson rings and point out the difference with the case of Baer radical rings. Suppose to the contrary that S contains an infinite descending chain of idempotents e\ > e^> Put Y = {ei, e2,...}. For n — 1, 2 , . . . , denote by Pn the ring of polynomials in n commuting variables over C. Let P^, be the ring of polynomials in commuting variables x\,X2,... over C. It is known that all Pn are Jacobson rings, but P^ is not Jacobson. Consider semigroup ring PooY. Put Rn = Pnen for n = 1,2,..., and Rs = 0 for s G S\Y. Then R is an S-graded ring. If G is a subgroup of S, then RQ = 0 or RQ = Pn for some n. Therefore R is a Jacobson ring. However, Poo is a homomorphic image of R. This contradiction completes the proof. For the case of Baer radical rings it suffices to replace each P„ by the quotient ring Rn/In, where Rn is the ring of polynomials in n commuting variables over C without free terms, and In is the ideal generated in P„ by all products x^Xj where 1 < i < j < n. • Lemma 6.21 If a ring R is the union of an ascending chain of Jacobson ideals R^, where \i < T, then R is a Jacobson ring. Proof. Evidently, J(R) = U / i < r i 7(P M ) and B{R) = U M