Multiplication rings and graded rings

This article was downloaded by: [University of Regina]On: 24 August 2014, At: 23:28Publisher: Taylor & FrancisInforma Ltd Registered in England and Wales Registered Number: 1072954 Registered office: MortimerHouse, 37-41 Mortimer Street, London W1T 3JH, UK

Communications in AlgebraPublication details, including instructions for authors and subscription information:http://www.tandfonline.com/loi/lagb20

Multiplication rings and graded ringsJosé Escoriza a & Bias Torrecillas aa Departamento de Algebra y Análisis Matemático , Universidad de Almería ,Almeria, 04120, SpainPublished online: 05 Jul 2007.

To cite this article: José Escoriza & Bias Torrecillas (1999) Multiplication rings and graded rings, Communications inAlgebra, 27:12, 6213-6232, DOI: 10.1080/00927879908826818

To link to this article: http://dx.doi.org/10.1080/00927879908826818

PLEASE SCROLL DOWN FOR ARTICLE

Taylor & Francis makes every effort to ensure the accuracy of all the information (the “Content”)contained in the publications on our platform. However, Taylor & Francis, our agents, and our licensorsmake no representations or warranties whatsoever as to the accuracy, completeness, or suitabilityfor any purpose of the Content. Any opinions and views expressed in this publication are the opinionsand views of the authors, and are not the views of or endorsed by Taylor & Francis. The accuracy ofthe Content should not be relied upon and should be independently verified with primary sources ofinformation. Taylor and Francis shall not be liable for any losses, actions, claims, proceedings, demands,costs, expenses, damages, and other liabilities whatsoever or howsoever caused arising directly orindirectly in connection with, in relation to or arising out of the use of the Content.

This article may be used for research, teaching, and private study purposes. Any substantial orsystematic reproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distributionin any form to anyone is expressly forbidden. Terms & Conditions of access and use can be found athttp://www.tandfonline.com/page/terms-and-conditions

http://www.tandfonline.com/loi/lagb20

http://www.tandfonline.com/action/showCitFormats?doi=10.1080/00927879908826818

http://dx.doi.org/10.1080/00927879908826818

http://www.tandfonline.com/page/terms-and-conditions

COMMUNICATIONS IN ALGEBRA, 27(1 Z), 62 13-6232 (1 999)

Multiplication Rings and Graded

Rings

Josg Escoriza Blas Torrecillas

Departamento de Algebra y Analisis MatemAtico

Universidad de Almeria

04120. Almeria. Spain

Introduction

Recently. many authors have been interested in m~iltiplication modules and

rings ( cf. 171, [8], [16], [20] and their references ). Hereditary rings, von Neu-

mann regular rings and special primary rings ( SPIR's ) are some examples of

this important class of rings. In fact, in the noetherian case, they are finite

direct sums of Dedekind domains and SPIR's. Group rings which are multipli-

cation are characterized in [3] and several papers by Gilmer ( cf. [lo] ) . Many

authors have studied grading rings having some multiplicative ideal theory (

Copyright O 1999 by Marcel Dekker, Inc.

Dow

nloa

ded

by [

Uni

vers

ity o

f R

egin

a] a

t 23:

28 2

4 A

ugus

t 201

4

6214 ESCORIZA AND TORRECILLAS

cf. [4], [14], [22], etc ). Our main purpose is to study graded rings which are

n~ultiplication, hlultiplication graded modules have been already studied in

[9, Section 51. Therefore, these results will be used in order to obtain the main

aims of this paper, which are to determine when a ring graded is mult~iplication

and to characterize mult,iplicat~ion graded rings ( gr-multiplication rings ) .

The organization of the paper is as follows: Section 1 recollects the nota-

tion and the elemental properties about graded modules and rings that we will

use in this paper. Section 2 is devot,ed to the study of multiplication graded

rings. After showing some families of gr-multiplication rings, some local char-

acterizations of them are achieved and a graded version of Mott's theorenl is

obtained. Sometimes, the arithmetical structure is determined by the gradua-

tion of the ring ( see, for example, [4], [5], [6], [18] ). The aim of Section 3 is

to study the relations between the following properties: R is gr-multiplication

and R is multiplicat~ion in order to find sufficient. ~ondit~ions for a graded ring

t,o be a mult,iplicat,ion ring. In the finite case, we characterize commut,at,ive

twisted group algebras which are multiplication when the ground field is per-

fect. Moreover, a structure theorem is obtained in the noetherian case when

the grading group is torsion-free.

1 Preliminaries.

Recall from [18] the basic definitions on graded rings and modules. A

ring R graded by a group G will be denoted by R = $gEGRg, where R, is an

Dow

nloa

ded

by [

Uni

vers

ity o

f R

egin

a] a

t 23:

28 2

4 A

ugus

t 201

4

MULTIPLICATION RINGS AND GRADED RINGS 6215

additive subgroup of R and R,. R h C Rgh for every g, h in G. If the inclusion is

an equality, t,hen the ring is called strongly graded. If an element of R belongs

to UgEGRg = h(R), then i t is called homogeneous. An R-module M is said

to be a graded module if M = $gEGMg for a family of subgroups { M g ) g E G of

A4 such t,hat Mh. R , 2 Mhg for every g, h in G . In a similar way, we define

a strongly graded module. A graded submodule N of M ( N s9, M ) is a

submodule verifying N = e g E G ( N n M,). We write L g T ( M ) for the lattice of

all graded submodules of M . The identity element of G is denoted by e. R

represents a commutative ring with an identity element and t,he category of

right R-modules graded by a group G and degree preserving homomorphisms

is denoted by gr-R.

If hl is an R-module and N is its submodule, the ideal {r E R ; M . r N )

will be denoted by (N : M ) . If Ad and N are graded, then so is (N : M ) (

cf. [9, Lemma 5.11 ) . 4 n ideal is gr-maximal if it is maximal in the lattice

of graded ideals. The graded Ja.cobson radical of a ring R is defined as the

intersection of all gr-maximal ideals. It will be denoted by JgT(R). A graded

ring having an only gr-maximal ideal is called graded local ( gr-local ). If P is

a graded ideal of a graded ring R , P is called graded prim.e if it verifies that

I. J E P implies that I C P or J E P for any graded ideals I, J of R . The

set of all graded prime ideals is denoted by SpecgT(R). The set of gr-maximal

ideals will be denoted by M a x g T ( R ) . Recall that a graded ring is gr-noetherian

if every graded ideal is finitely generated.

Dow

nloa

ded

by [

Uni

vers

ity o

f R

egin

a] a

t 23:

28 2

4 A

ugus

t 201

4


Given a ring R graded by G and a multiplicative subset S h(R), the ring

of fractions S-IR turns into a ring graded by G by means of ( S P ' R ) ~ = {S; s E

S , r E R, g = (deg s)- ldeg r ) for every g E G, where deg s represents the

degree of s. Moreover, in the strongly graded case, S-'R is also strongly graded

( see [lt?, Proposition A.I.6.2 and Corollary A.I.6.31 ). When S = h ( R ) - P

for some graded prime ideal P of R, we will write R$ for S-'R. Recall that

S-'M can be defined as S-'R B R M , when M is an R-module. Throughout

this paper, unless otherwise stated, every graded ring is non-trivially graded.

2 Multiplication graded rings

Firstly, we will show that there exist multiplication graded rings that are

not multiplication rings. Then, we will give a local characterization of gr-

nlultiplication rings.

Let R be a graded ring. An R-module M is a multipl%cation graded module

( gr-multiplication module ) if for every graded submodule N , there exists an

ideal I of R such that N = M I . This ideal can be taken graded ( see 191 ) .

Definition 2.1 A graded ring R i s said to be a gr-multiplication ring if every

graded ideal i s gr-multiplication as a n R-module.

I t is obvious that every graded ring which is multiplication is gr-multipli-

cation. Recall that a graded Dedekind domain ( gr-Dedekind domain ) is an

integral domain in which every non-zero graded ideal is invertible ( cf. [18]

Dow

nloa

ded

by [

Uni

vers

ity o

f R

egin

a] a

t 23:

28 2

4 A

ugus

t 201

4


and its references ). Multiplication graded domains have been characterized,

in an analogous way to the ungraded case in [9, Proposition 5.61, as being just

gr-Dedekind domains.

Remark. Let R be a Dedekind domain. In [21], i t is proved that if R is a

Dedekind domain, then R [ Z ] is a gr-Dedekind domain. Therefore, the group

ring R [ Z ] will be a gr-Dedekind domain and hence, it is a gr-multiplication

ring. However, if R is not a field, then it is not a multiplication domain because

it is not a Dedekind domain ( see [lo, Theorem 13.81 ).

Anot,her kind of gr-multiplication rings consists of all graded von Neumann

regular rings or, more briefly, gr-regular rings ( they have been defined in [23]

) . .4 ring R is gr-regular if for any homogeneous element x of R, there exists

y E R such that x = x y x . The group ring (IIg,Ri)[ZZ], where Ri = 222, is

gr-multiplication ( it is gr-regular) but not multiplication ( cf. [9] ). Thus

there are gr-multiplication rings that are not multiplication rings.

Now, we will characterize the gr-noetherian gr-multiplication rings by local-

ization. Before that, we recall the graded version of discrete valuation domain

( gr-DVD ) . If K is a graded field, a graded subring of K, R , is a discrete

graded-valuatzon domain ( gr-DVD, see [18, p. 1641 ) if for any homogeneous

element x of K , it holds that x or x-I belongs to R, B being its valuation

group. We define the graded version of a special primary ring. A graded ring

is said to be a graded special primary ring ( gr-SPIR ) if it is gr-local and every

Dow

nloa

ded

by [

Uni

vers

ity o

f R

egin

a] a

t 23:

28 2

4 A

ugus

t 201

4


proper graded ideal is a power of its only gr-maximal ideal. Then, it holds the

following easy consequence of [9, Theorem .5.4].

Proposi t ion 2.2 If R i s a gr-noeth,erian, t h e n R i s gr-multiplication if and

only zf R$ i s a gr-multiplzcation ring for every P E S p e c g T ( R ) .

Lemma 2.3 If R is a gr-multiplication r ing and P i s a n y gr-ma,dm,al ideal

of R , t hen between Pn+l and Pn there i s n o graded ideal strict ly contained for

every possiti~ve integer n.

Proof: Suppose that A is a graded ideal such that Pn+l C A Pn. Since

R is gr-multiplication, there exists a graded ideal B such that A = BPn. If

B C P , then A C Pntl and therefore, A = P n t l . If not, B + P = R and

hence, Pn = Pn(B + P ) = pn+' + PnB = A. 0

The next result is a local characterization of gr-multiplication rings similar

to the achieved one for the ungraded case.

Theorem 2.4 A gr-noetherian ring £2 graded by a n ordered group 2s gr-mul-

tiplication i j and only if R: i s a g r -SPIR for every P E SpecgT(R) .

Proof: Firstly, we shall prove that every gr-local gr-multiplicat,ion gr-noe-

therian ring graded by an ordered group is a gr-SPIR. Let) R be a gr-local

gr-multiplication ring. If R is a domain, then it is a gr-Dedekind domain by

[9, Proposition 5.61. Now, [18, Theorem 11.2.11, implies that R is a gr-DVD

and, therefore, a gr-SPIR. Now, suppose that R is not a domain. By [9,

Dow

nloa

ded

by [

Uni

vers

ity o

f R

egin

a] a

t 23:

28 2

4 A

ugus

t 201

4


Proposition 5.181, the only gr-maximal ideal, P, can be written as P = (z) for

some z E h ( R ) . Since R is not a domain and G is an ordered group, by 118,

Proposition A.II.l.41, there exist some homogenous zero divisors in R. Now, i t

will be proved that if a E h ( R ) is a zero divisor, then there exist a unit u E R

and a possitive integer m such that a = usm. In fact, since ( a ) E C F ( R ) ,

there exists a non zero element a1 E h ( R ) such that a = a l x . If a is a unit,

the process ends. If not, a1 E ( x ) and therefore, there exists another non zero

homogeneous element a2 verifying a1 = azx and a = azx2 . As before, if a2 is

a unit, we have finished. If not, we repeat the process. If the wished property

were not true, we would obtain an ascendent chain of graded ideals ( as their

generators are homogeneous ), ( a l ) E ( a n ) . . . E (a,) 5 . . .. Since R is gr-

noetherian, there exists 72 such that (a,) = (anr l ) . Then, a = anxn = a,+lznfl

w ~ t h o n = an+lx. I t holds that a, ,~ = a, b for some b E h ( R ) as a,+l E (a,). I t

IS deduced that a,, ,~ (1 - b x ) = 0 , but 1 - bx is a unit because ~t is homogeneous

and it does not belong to P It follows that n,+~ = 0 and , consequently, a = 0,

which is a contradiction. Thus there exist zero divisors, a = u x n and b = vx",

nb = 0 and u, u being units. It follows that xntm = 0. Therefore, PZ = 0 for

some poss~tive integer z . By [9, Lemma 5.221, L f l ( R ) is totally ordered under

inclusion. It suffices to apply Lemma 2.3 to finish the first part of the proof.

By Proposition 2.2, the converse is clear.

Recall that a ring is said to be a gr-PIR is every graded ideal is principal. A

graded ideal which is generated by a homogeneous element is called gr-cyclic.

Dow

nloa

ded

by [

Uni

vers

ity o

f R

egin

a] a

t 23:

28 2

4 A

ugus

t 201

4


In the general graded local case. we obtain the next result.

Proposi t ion 2.5 Every gr-local gr-mz~ltiplicntion. rin.g is gr-PIR.

Proof: Let R be a gr-local ring. Then, we shall prove that every gr-multipli-

cation ideal is gr-cyclic. Let P be the gr-maximal ideal of R . Suppose that .4

is a non-zero proper graded ideal of R . Since A is gr-multiplication, given 0 f

a E h ( A ) , a R = 4 B , for some B, Sg, R. Then, A = C a e A a R = C a t A A B a =

A(CatCA B,). If CaEA B, = R , then B, = R for some a E A because R is gr-

local. In this case, A = AR, = aR and hence, A is gr-cyrlic. If CatA Ba # R,

then A = AP. For 0 # a E h ( 4 ) , it holds nR = AB, = APB, = AB,P = a P .

By [9, Proposition 5.71, a = 0, a contradiction. 0

From 19, Theorem 5.31, we immediatly obtain the following characterization

of strongly gr-multiplication rings. The result can be also deduced from the

category equivalence between R-gr and Re-Mod in case R is strongly graded.

Proposi t ion 2.6 Let R be a strongly graded ring. Then, R is gr-multiplication

if and only i f Re is a multiplication ring.

As a consequence of Proposition 2.6, we can give another family of gr-mul-

tiplication rings. Recall from [22] that the generalized Rees ring associated to

I ( an invertible ideal ) is R ( I ) = CnEzXnIn , with the obvious ZZ-graduation.

It is well-known that these rings are strongly graded. By Proposition 2.6, if R

is multiplication, then R ( I ) is ZZ-gr-multiplication. Note that this generalizes

[22 , Theorem 2.31.

Dow

nloa

ded

by [

Uni

vers

ity o

f R

egin

a] a

t 23:

28 2

4 A

ugus

t 201

4

MULTIPLICATION RINGS AND GRADED RINGS 622 1

If R is a Dedekind domain, G is a finite group and t is a generalized 2-

cocycle, i. e. not necessarily taking its values in the units of R, then the

generalized twisted group ring (cf. [19] ), Rt[G], is gr-multiplication.

The next result can be considered as a Mott's theorem ( ef. [15] ) for graded

rings.

Theorem 2.7 Let R be a ring strongly graded by a n ordered group. Then, , R

is gr-multiplication i j and only i j every P E SpecgT(R) i s gr-multiplication.

Proof: Suppose that every graded prime is gr-multiplication. Let Q he a

prime ideal of Re. By [18. Lemma A.I.8.15 and Lemma A.II.2.21, there exists

P E Specgr(R) such that Pe = Q. By the assumption, P is gr-multiplication.

By [9, Theorem 5.31, Pe is multiplication as an Re-module. We have proved

that every prime ideal of Re is multiplication. By Mott's theorem, Re is

multiplication. By Proposition 2.6, R is gr-multiplication. Since the other

implication is obvious, the equivalence is proved.

3 Graded rings and multiplication rings

Now, our purpose is to det,ermine when a graded ring is a multiplication

ring. For this reason, we relate that R is gr-multiplication, that Re is a multi-

plication ring and that R is multiplication.

Recall some concepts which appear in the next result. Given a ring strongly

graded by G, R , and a subgroup H 5 G, it is well-known that the ring R ( ~ ) =

Dow

nloa

ded

by [

Uni

vers

ity o

f R

egin

a] a

t 23:

28 2

4 A

ugus

t 201

4


ElgEHR, is strongly graded ( cf [18, Example 1 3 1, p 151 Recall that

a commutative rmg 1s called rfduced or semzprzme rzng if ~t has no nilpotent

elements d~fferent from zero

Proposition 3.1 Let R be a ring graded by G an,d let W be a subgroup of G.

Then,

1. if R is gr-mzrltiplication rzng, then R, is multiplicntion~.

2. if R is strongly graded and R zs multiplication, so zs R ( ~ ) .

9. if R is noetherian a,n.d stron,gly graded and G is finite with order invertible

in the reduced ring R,, then R is multiplicntion if a,nd only if Re i,s

multiplication.

Proof: In order to prove 1, let A,, Be be ideals of Re with A, 5 Be. Consider

the graded ideals, A = R,4, and B = RB,. By hypothesis, A = RI for some

graded ideal I of R. 111 particular, their components of degree e have to be

equal. Thus A, = xgtG BgIg-1 = Be(xgEG RJg - I ) and therefore, the ideal B,

is multiplication and hence, Re is a multiplication ring.

Now, we shall prove 2. Every graded ideal of R ( ~ ) is generated by its

components of degree e as it is strongly graded. Suppose that R ( ~ ) Je R ( ~ ) I ,

with I, and J, ideals of Re. I t follows that J, C I, and therefore, RJ, RI,.

By hypothesis, RJ, = RAeIe for some ideal A, of Re. By taking direct sum in

H, it remains R ( ~ ) J, = R ( ~ ) A , I , = ( R ( ~ ) A , ) ( R ( ~ ) I , ) .

Dow

nloa

ded

by [

Uni

vers

ity o

f R

egin

a] a

t 23:

28 2

4 A

ugus

t 201

4


To prove 3, suppose that Re is multiplication. By [18, Exercise C.I.6 (3)],

R is reduced. By [ll, Corollary 21, Re is hereditary. By [17, Corollary 2.121,

so is R and, hence, it is multiplication. The converse holds by 1.

Remarks. z) The implication which appears in Proposition 3.1 (1 ) is strict.

Consider the polynomial ring R = ZZ[X] graded by ZZ in the usual way. By

[18, Remark B.II.2.91, it is a domain but it is not a gr-Dedekind domain. By [9,

Proposition 5.61, R is not gr-multiplication. However, its component of degree

zero, 23, is clearly multiplication.

71) Proposition 3.1 (3) gives us that the above defined generalized twisted

group ring Rt[G] is multiplication when the order of G is invertible in R.

Some results have been obtained in the finite case. In general, it is not

always true that if R is gr-multiplication, then it is multiplication. For exam-

ple, the group ring R = Zs[Z2 $ iZ3] is gr-multiplication because so is z6.

However, [lo, Theorem 19.101 gives us that R is not multiplication.

When the order of the group G is not invertible in R, the situation is

more complicated. First, recall the construction of twisted group rings. Let

@ . G --- Aut(R) be a morphism from the group G to that of automorphisms

of a ring R. Denote @ ( g ) = erg. A 2-cocycle is a mapping t : G x G --i U Z ( R )

( group of units of the center of R ) verifying t(g, h)t(gh, I ) = t (h , l ) % ( g , h l ) ,

where t ( h , l ) a g is t (h , I ) in the commutative case ( cf. [18, p. 261 ). In these

circumstances, the free R-module with base G and whose product is defined

Dow

nloa

ded

by [

Uni

vers

ity o

f R

egin

a] a

t 23:

28 2

4 A

ugus

t 201

4


as follows: ( C g E ~ a g ~ ) ( & G bhh) C i a cilr where = Cgh=l agbtgt(g, h)r

is called crossed product of the ring R and the group G . It will he denoted

hy R * G and it is graded in the following way: (R * G ) g = {rg; r E R}. If

the action is trivial, t,hen the consequential concept is t,hat of t,wist,ed group

ring with respect to t , which will be denoted by Rt[G] ( these concepts can be

found in [18] ) . We write Gp for the p-primary component of the finite group

G. When the ground ring is a field, we have the next result.

Proposit ion 3.2 Let Kt[G] be a commutative twisted group algebra, K being

a field and G being a finite group. If char(K) = 0 or char(K) = p and G p is

cyclic, then Kt[G] is multiplication. Moreover, if h' is perfect the converse is

also true.

Proof:

Suppose that char(K) = 0. Then R = Kt[G] is multiplication from Propo-

sition 3.1 (3). Let p = char(K) > 0. If p does not divide the order of G ,

then by Maschke's Theorem, Kt[G] is semisimple and therefore, it is multi-

plication. Suppose that p divides the order of G and G, is cyclic. B y [12,

Theorem 111.2.81, R is isomorphic to a quotient of a group algebra K [ H ] where

H has the same p-primary component as G. By [lo, Theorem lg.lO], K [ H ] is

multiplication and, therefore, R is multiplication as well.

Conversely, suppose that Kt[G] is multiplication and char(K) = p > 0,

K being a perfect field. By Proposition 3.1 (2), Kt[G,] is also multiplication.

Dow

nloa

ded

by [

Uni

vers

ity o

f R

egin

a] a

t 23:

28 2

4 A

ugus

t 201

4


Since K is perfect, we can apply [12, Lemma V.8.131 to obtain Kt[G,] K[G,].

By [lo. Theorem 19.101, G, is cyclic.

Remarks.2) If K is not perfect, t'hen the result is not always true. In fact, it,

suffices to consider R = Z Z ( X ) ~ [ Z ~ @ Z*] where Z 2 ( X ) is the field of rational

f1inct,ions with coeficients in Z 2 and t verifies that its restriction to Z2 is not

cohoundary. Then, Zz(X)t [Za] is a field ( see [12, p. 2531 ) and, by the

first part of Proposition 3.2, R is multiplication ( the restriction o f t has been

denoted by t as well ) .

ii) There are examples of twisted group rings and t,herefore, st,rongly graded

1,y a finite group, which are multiplication or not depending on the cocycle.

For instance, the twisted group ring ZZi[C4] where the 2-cocycle t is defined

as follows: t(O,O).t( l .O). t( l , 1 ) , t (3 ,O) , t (2 ,0 ) and t ( 2 , l ) are equal to 1, and

t(3,3) = 3 = t (3 ,2) = t (3, 1) = t (2.2). Of course, t is symmetrical. This

ring is clearly isomorphic to Z ~ [ , Y ] / ( X ~ - 3 ) , which is a SPIR ( see [5 ] ) and,

therefore, it is a multiplicat~ion ring. However, if we consider the graded ring

R = Z;[C4] where now, the cocycle t is the convenient one to establish the

isomorphism with Za[X]/(.Y4 - 5 ) , then R is not multiplication. Accordingly.

it seems interesting to study the influence of the cocycle in order to obtain a

nlultiplication ring and we will deal wit,h this problem in a subsequent paper.

Henceforth our work will be focussed on graded ring whose grading group is

torsion-free.

First. the local case will be studied

Dow

nloa

ded

by [

Uni

vers

ity o

f R

egin

a] a

t 23:

28 2

4 A

ugus

t 201

4


Lemma 3.3 If R is a local rin,g strongly graded by Z. t hen R is a, grolrp rin.g.

Proof: Firstly, it will he proved that if R is a graded ring which is local, then

R, is local and therefore, R is gr-local. Assume that P is the only maxinlal

ideal of R and that there is two different maxirnal ideals PI and 4 of Re.

Then, PI + Pz = R,. The graded ideals RP1 and RP? are different hecause

so are their components of degree c and they are proper ideals because their

components of degree e are different from Re. Thus, RP1. RP? 5 P . It holds

R, = PI + P2 P n Re. Hence. R = RR, 2 P , which is a contradiction. The

equivalence between that Re is local and that R, is gr-local give it to us [2,

Proposition 21.

Since R is local, Re is local. By [IS. Corollary A.I.3.261, R is a crossed

product. By [18, Corollary .\.I.3.24]. R is a group ring, with ground ring Re.

D

If R is a domain strongly graded by Z, it is obtained the following structure

theorem.

Theorem 3.4 Let R be an in,tegraE d o n ~ a i n stron.gly graded by Z. Then . R is

mult iplication 7i an,d only if R is a group ring u h o s e ground ri.n,g is a field. i.

e., R 2 K [ X , X - ' 1 . K being a field.

Proof: If R is a multiplication domain, then it is a Dedekind domain and, in

part,icular, it is a Priifer domain. By 16, Lemma 4.21. R K[Z], where K is a

field. It follows from [lo, Theorem 13.81 that K [ Z ] is a Dedekind domain. 0

Dow

nloa

ded

by [

Uni

vers

ity o

f R

egin

a] a

t 23:

28 2

4 A

ugus

t 201

4


The next result extends the case of finite direct sums of P.I.D,'s and

S P.I.R.'s which appears in [ 5 , Proposition 11 and it is necessary to prove

the following structure theorem.

Lemma 3.5 Let R be a noetherian midtiplication rin,g ( stron,gly ) graded b y

a torsion-free group G . Then,, R is finite direct sum of Dedekin,d domains and

special prim,ary rings, each of t h e m being ( strongly ) graded by G .

Theorem 3.6 Let R be n n o ~ t h e n n n rzng strongly graded by 22. T h e n , R as

rnult~plzcatton zf and only ~f Re 2s a finzte product of fields Moreorier, zn thzs

c a w . R I S a y o u p rmg .

Proof: Firstly, suppose that R is multiplication. Then, R - Dl $ . . . @ D, @

Sl @ . . . 9 St where Di ( 1 5 i < r) is a Dedekind domain and ,', (1 5 j 5 t )

is a special primary ring ( SPIR ) . By [5> Proposition 11, so are the Di's

and the Sj's are strongly graded by G. The degree e component of R is

R, = ( D l ) , e . . . (D,),B ( S I ) , $ . . . $ (St),. By Proposition 3.4, (D i ) , is a field

for 1 < i _< r . If S is a SPIR, it is local. If, moreover, it is strongly graded, by

Lemma 3.3, it is group ring. However, S [ Z ] cannot be local by [ l o , Corollar!-

19.21. It follotvs that Re is a finite direct sum of fields.

Xow, suppose that Re is a finite direct product of fields. By [18, Corollary

1.3.261. R is a crossed product of Re and, by [18, Corollary 1.3.241, R 2 R , [ Z ] .

By 110, Theorem 38.101, it is deduced that R is mult,iplication. C]

In the following case, an equivalence between being gr-multiplication and

being n~ultiplication is achieved

Dow

nloa

ded

by [

Uni

vers

ity o

f R

egin

a] a

t 23:

28 2

4 A

ugus

t 201

4


T h e o r e m 3.7 If R i s a gr-noeth,erian ring possitively graded b y Z, t hen R is

gr-multiplication zf and only if i t i s m.ultip1ication.

Proof : In the first place, we claim that if R is gr-noetherian and gr-multipli-

cation ( for any group G )> then every non-zero graded ideal is a finite product'

of graded prime ideals. Let A &, R. If -4 E Specgr (R) , we have finished. If

not, there exists P E M a x g T ( R ) such that A P. Since R is gr-multiplication,

.4 = PB, where B = (,4 : P), which is a graded ideal of R containing ,4. If

B E Spec"(R), the process finishes. If not, we can'repeat the above reasoning.

It is obtained an ascendent chain of graded ideals A E B C C . . .. Because R

is gr-noetherian, the process ends and therefore, A can be written as a finite

product of graded prime ideals.

Suppose that R is gr-multiplication. By the first part, every non-zero

graded ideal of R is a product of graded prime ideals. By [I, Theorem 71,

every non-zero ideal ( graded or not ) is a finite product of prime ideals. By

[13, Theorem 9.101, R is a finite direct sum of Dedekind domains and SPIR's

and hence, it is multiplication. The other implication is always t,rue.

If the ring is graded ( not necessarily strongly graded ) by a torsion-free

group, noetherian multiplication rings can also be characterized in the follow-

ing way.

T h e o r e m 3.8 Every noetherian multzplzcation ring graded b y a torsion-free

group, i s a finite direct s u m of triuially graded Dedekind dom.a,ins, trivially

Dow

nloa

ded

by [

Uni

vers

ity o

f R

egin

a] a

t 23:

28 2

4 A

ugus

t 201

4


graded SPIR's and rings of the fornz K[X], K1[ZZ] and K"[X]/(Xn) with n 1

2, K , K' and K" being fields.

Proof: Since R is n~ultiplication and noetherian, it is a direct sum of a finite

number of Dedekind domains and SPIR's, which will be graded by the same

group G. By [4, Corollary 51, domains which are not trivially graded will be

of the form K [S] or K [ Z ] for some field K . By [5, Proposition 31, the non-

trivially graded SPIR's are of t,he form K[X]/(Xn). It suffices to join these

results to deduce the theorem. 0

Acknowledgements

The two authors are most grateful to professor Fred van Oystaeyen for his

suggestions and ideas. We have been partially supported by grant PB95-1068

from DGES and grant CRG 971543 from NATO.

References

[I] Anderson, D. D. and Matijevic, J., Graded x-rings, Can. J . Math., no.3,

(1979) 449-457.

[2] Beattie, M., DBsciilescu, S. and ?J&st&sescu, C., A note on semilocal graded

rings, Rev. Roumaine Math. Pures Appl., 40 3-4, (1995) 253-258.

[3] Chouinard, L. G., Hardy, B. R. and Shores, T. S., Arithmetical and Semi-

Dow

nloa

ded

by [

Uni

vers

ity o

f R

egin

a] a

t 23:

28 2

4 A

ugus

t 201

4


hereditary Rings, Comm. in Algebra, 8 (17), (1980) 1593-1652.

[4] Decruynaere, F. and Jespers, E., Dedekind donzains and graded rings,

Proceedings of the Amer. Math. Soc., 110 (I ) , (1990) 21-26.

151 Decruynaere, F. and Jespers, E,., Graded commutatitre principal ideal

rings, Bull. Soc. Math. Belg., 43 (2), Ser. B, (1991) 143-150.

[6] Decruynaere, F. and Jespers, E., Priifer domains and graded rings, J.

Algebra, 150, (1992) 308-320.

171 Erdogdu, V., Regular multiplication rings, J , of Pure and Applied Algebra,

59, (1989) 55-59.

[8] Erdogdu, V., Multiplication modules which are distributive, J. Pure and

Applied Algebra, 54, (1988) 209-213.

[9] Escoriza, J . and Torrecillas, B., Multiplication Objects in Conmutatiue

Grothendieck Categories, Comm. in Algebra, 26 (6), (1998) 1867-1883.

[lo] R. W. Gilmer, Commutative Semigroup Rings, The University of Chicago

Press, Chicago (1984).

[ll] M. Griffin, Multiplication rings via their total quotient rings, Can. J .

Math. Vol. XXVI (2), (1974) 430-449.

[12] G. Karpilovsky, Projective representations offinite groups, Marcel Dekker,

New York (1985).

Dow

nloa

ded

by [

Uni

vers

ity o

f R

egin

a] a

t 23:

28 2

4 A

ugus

t 201

4

MULTIPLICATION RINGS AND GRADED RINGS 623 1

[13] M. D. Larsen and P. J. McCarthy, Multiplicative theory of ideals, Aca-

demic Press, New York (1971).

[14] Le Bruyn, L., Van den Bergh, M. and Van Oysbaeyen, F., Graded Orders,

Birkhauser, Boston (1988).

[15] hlott,, J . L., Equivalent conditions for a ring to be a multiplication ring!

Can. J , Math., 16, (1964) 429-434.

[16] Yaoum, A. G. and Al-Alwan, F. H., Dense submodules of multiplication,

modules, Comm. in Algebra, 24 (2), (1996) 413-424.

1171 NBstasescu, C., Strongly graded rings of finite groups, Comm. in Algebra.

11 (lo), (1983) 1033-1071.

[18] NCst&sescu, C. and Van Oystaeyen, F.,Graded rings theory, North-

Holland, Amsterdam (1982).

[19] Nauwelaerts, E. and Van Oystaeyen, F., Generalized twisted group rings

of finite groups. Prepint.

[20] Smith, P. F., Multiplication modules and projective modules, Periodica

Math. Hungarica, 29 (2), (1994) 163-168.

[21] Vm Oystaeyen, F . and Van Deuren, J . P., i lri thmetically graded rings,

Leetures Notes in Math., 825, Ring Theory ( Antwerp 1980 ), Proceed-

ings, (1980) 279-284.

Dow

nloa

ded

by [

Uni

vers

ity o

f R

egin

a] a

t 23:

28 2

4 A

ugus

t 201

4


(221 Van Oystaeyen, F., Generalzzed Rees Rmgs and Arzthmetzcal Graded

Rings, J . of Algebra, 82, (1983).

[23] Van Oystaeyen, F., Note on graded won Net~mann regular. rings, Rev.

Roumaine Math. Pures Appl., 29 (3) , (1984) 263-271.

Received: July 1998

Dow

nloa

ded

by [

Uni

vers

ity o

f R

egin

a] a

t 23:

28 2

4 A

ugus

t 201

4

Documents

Multiplication rings and graded rings