Internat. d. Math. & Math. Sci. VOL. 18 NO. 2 (1995) 311-316

311

### CLASSICAL

QUOTIENTRINGS OF### GENERALIZED

### MATRIX

### RINGS

DAVID G. POOLE

DepartmentofMathexnatics, TrentUniversity

Peterborough,Ontario, Canada,K9J 7B8

e-mail: DPOOLE@TrentU.ca

and

PATRICK N.STEWART

### Department

ofMathematics, Statistics andComputingScience Dalhousie University,Halifax,NovaScotia,Canada,B3H### a

J5e-mail: stewart@cs.dal.ca

### (Received

March18,### 1993)

ABSTRACT.

### An

associativering### R

withidentity is ageneralizedmatrix ringwithidempotentset

### E

if### E

is afinite set oforthogonal idempotents of### R

whose sumis 1. Weshow that,inthepresence of certain annihilatorconditions,such aringissemiprimerightGoldie ifand onlyif

### ere

issemiprimerightGoldie for alleE### E,

andwecalculate theclassicalright quotient ring of### R.

### KEY WORDS AND

PHRASES. Generalizedmatrixring, quotientring, Goldieconditions.1991AMS SUBJECT CLASSIFICATION CODES. 16P60, 16U20.

1. INTRODUCTION.

Each ring considered in this paper is associative and has an identity. Such a ring

### R

is ageneralizedmatrix ring with idempotent set

### E

if### E

is afinite setoforthogonalidempotents of### R

whosesumis 1.In this paper, we show that, in the presence of certain non-degeneracy conditions, a generalizedmatrix ring

### R

withidempotent set### E

is semiprime right Goldie if andonly if### eRe

issemiprime right Goldie for all eE

### E,

and we calculate the classical right quotient ring of### R.

### Kerr’s

example### [4]

of a right Goldie ring whosematrix ring is not right Goldie shows that oursemiprimeness conditioncannotbeomitted.

Examples ofgeneralized matrix rings include incidence algebras of directed graphs with finite number of vertices

### (see

### [5]

and### [9]),

structural matrix rings### (see

Van Wyk### [13]

and subsequent### papers),

endomorphism rings of finite direct sums of modules and Morita context rings. Sands### [10]

observed that if### [S,V, W,T]

isaMoritacontext,thenis aring. TheseMorita context ringsarepreciselygeneralized matrixringswith idempotentsets

312 D. G. POOLE AND P. N. STEWART Miiller’s computation of the maximal quotient ring in

### [8].

A generalized matrix ring R with idempotent set E is called a piecewise domain if for all

e,f, gEE,

### :rEeRf

and yGfR9, wehave,ry=0implies z=0or_{9=0.}

These rings have been
studied insonedetail- see, for instance, ### [2]

and### [3].

Wedenote the1)rime radicalofaring R 1)y p(R)and if and

_{f}

areidempotents ofR, _{7}

f,
p(eRf)denotesthe set ### {a’e

### eRf:xfRe

C_p(### Re)}.

PROPOSITION 1.1

### (Sands

### [10]).

IfR isa generalizedmatrix ring with idempotent set E, thenp(R)=

### _,

p(eR### f).

e,fEPROOF. If

_{EI}

there is nothingto prove andif _{EI}

2 this is Theorem in Sands
### [10].

Assumenow that_{EI}

-n### >

2 and that the theoremis truefor generalized matrix ringswith idempotent sets of cardinality less than n. Let e

### E

and set### E=

### {e,

### 1-e}.

Then### R

is ageneralizedmatrixringwithidempotentset

### E

andsoSands’ result implies that p(R) p(eRe)+### p(eR(1-e))

### +

### p((1-

### e)Re)+

p((1-### e)R(1 -e)).

Since

_{(1- e)R(1-e)}

is ageneralized matrix ringwith idempotent set E ### E\{e},

our inductionhypothesis implies that

### ((

### )n(

### ))

### (In).

### f,

gE### E

Also, it is clear that

### p(eR(1-e))=

### p(eRf)and

### p((1-e)Re)=

### p(fRe),

so the resultfollows. !t E IeE [_]

Let

### R

be a generalized matrix ring with idempotent set E. We say that the pair### (R,E)

satisfiesthe_{left}

(respectively, right) annihilator conditionif for all e,_{f}

### E,

0 x### e

### err

implies that_{xfRe}

0 (respectively, ### fRex

### 7

### 0).

This concept is defined in### [12]

where right and leftareinterchanged.

### COROLLARY

1.2.### (Wauters

and### Jepers

### [12]).

The following conditions on a generalizedmatrixringwithidempotentset

### E

areequivalent.### (a)

### R

issemiprime.### (b)

### (R,E)

satisfies the left annihilator condition and### ere

is semiprime for all e E.### (c)

### (R,E)

satisfiestheright annihilator conditionand### ere

issemiprimefor alle### E.

Ci2.

### THE

GOLDIE### CONDITIONS.

The right singular ideal of a ring S will be denoted by

### Z(S),

and the right singular submodule ofaright S-module### M

willbedenotedby### Z(M).

### So,

if### R

is ageneralizedmatrixringwith idempotent set

### E

and e,_{f}

### e

### E

withe f,then### Z(eRe)

isthe right singularidealof the ring eReand### Z(eRf)is

the rightsingularsubmodule of theright_{fRf-module}

### err.

PROPOSITION2.1. Let

### R

beageneralizedmatrixringwithidempotentset Eand supposethat

### (R,

### E)

satisfies the leftannihilator condition. Then### Z(R)

### Z(Rf).

### e,fE

PROOF. Lete,

_{f}

E### E

andsupposethat x### Z(R).

Then_{ezf}

### Z(R),

sothereis anessentialrightideal

### I

of### R

such that_{exfI}

0. Toshowthat_{ezf}

### e Z(eRf)

it suffices tosho,that_{fir}

is
CLASSICAL QUOTIENT RINGS OF GEN1;RAI,I_ZED NATR1X RINGS 313

### IOAFtO.

L(’t_{OuEIClAR.}

SinccINARisarightidealthereisanidempotent _{9EEsuch}

that 0

### y

ugEINAR, and ug .lug 1)(’cans(, ugE### AR

C__{fRfR.}

Since (R,E) satisfies the left
annihilatorcondition, 0

_{#}

(.fug)gRf C_(I AR)fl### fRf

C__{fRf}

CA. It follows that
Z(R)C_ Z(cR.f). ,.fEE

Conversely, suppose that e,fEE and egfE

### Z(eRf).

Then_{9H}

0 hn" some essential
right dcal H of

_{fRf.}

Let J ### {rE

R:.f,’E### HR}.

Clearly, J is a right ideal of R and9,!=egfJ=(eyf)fJC_(egf)HR=O, so to show that 9EZ(R)it is enough to show that J is

essential in R. Let

### B

be a nonzero right ideal ofR. IffB=0, then B_CJ and so_{BClJ0.}

Nowassume

_{fB}

## -

O. ThenfBg### #

0 forsom,_{9}EE, andsotheleftannihilator conditionimplies

that

_{fBf}

_{7}

O. So we seethat _{fBf}

is anonzero right ideal of_{fRf.}

Thus _{fBf}

cl### H

### #

0 and soBVIJ

_{#}

0becauseHC_HR. 13
COROLLARY 2.2. If

### R

isageneralizednatrix ringwithidempotentset### E

such that### (R, E)

satisfies the left annihilator condition, then

### R

isnonsingulaxifand only ifeRe is nonsingular foralleEE.

PROOF. In view of the proposition, we need only show that

### Z(R)

0 implies that### Z(eRe)

0for some eEE.### Suppose

that 0### y

xE### Z(R).

Then 0_{7}

### exf

E### Z(R)

for some### e,f

E.Theright annihilatorconditionimplies that

### (ezf)fRe

0 andso### ere

Cl### Z(R)

O. Itnowfollowsfromthe proposition that

### Z(eRe)

O. I-iThe right uniform dimensionofa ringR (respectively, right R-module

### M)

will be denotedby

### d(R)

(respectively,### d(i)).

PROPOSITION 2.3. Let

### R

be a generalizedmatrix ringwith idempotent set### E

such that### (R,E)

satisfies the left annihilator condition. If### d(R)<

o0 then### d(eRe)<

e for all eEE.### Moreover,

if### R

is semiprime and### d(eRe)<

ocfor alleE### E,

then_{d(eRf)<}

cx for all### e,f

E### E

and hence### d(R) <

cx.PROOF. Assume that

### d(R) <

cx,eE### E

and### ]A,

is a direct sum ofnonzero right idealsof### ere.

To prove that### d(eRe)<

oc it is enough to show that### ]A,R

is direct, and to accomplishthisweneedonlyshow that ]

### A,Rf

isdirect for each_{f}

EE. ### Suppose

that_{f}

E### E

and### b,

E### A,Rf

are such that ]### b,

0. Since### b,fRe

C_### A,

and ]A, is direct,### b,fRe

0 for all i. Thus the leftannihilator conditionimpliesthat

### b,

0 for all and hence ]### A,Rf

is direct.Now assumethat

### d(eRe)

### <

0for all eE### E

andsupposethat ]### N,

isadirect sumofnonzero### fRf-submodules

of### eRf.

Since 0_{7}

### N,

C_### err

the left annihilator condition implies that### N,fRe

### #

0, and each_{NfRe}

is a right idealofeRe. Let ### K

_{NffRefl}

_{2}

### {N,fRe:i

### #

j}. Then### KeRr

C__{N,}

Ci ### {N,:i

_{7}

j} and so ### KeRr

0. Since If C_KeRf,### K

0 and so### K

0 because eRe is semiprime by Corollary 2. Thus_{Z}

### N,fRe

is direct and so### d(eRf)<_

### d(eRe).

It follows that### R

hasfiniterightuniform dimensionas aright### ( 2

### eRe-module

andsocertainly_{d(R)}

### <

odi3From Corollary 2, Corollary4andProposition5we

### otan

thefollowingtheorem.THEOIM 2.4. Let

### R

be a generalized matrix ring with idempotent set E. If### R

is semiprimeright Goldie, then so too axethe rings### eRe,

eEE. Conversely, if### (R,E)

satisfies the left annihilator condition and### ere

is semiprime right Goldie for all e### E,

then### R

is semiprime right Goldie.3.

### THE QUOTIENT

R/NG.314 _{D.G.} _{t’OOLE AND} _{P.} _{N.} _{STEWART}

bimodule Ore condt,onif for each

### ,

E :!an,leach regular ’lemcnt### c

Sthereisan_{m}

### M

andaregular

### c

Tsuch that### PROPOSITION

3.1. If### R

is a scmiprime right Goldie generalized matrix ring withid’mpotentset

### E.

tlwn_{R."}

satisfies the right ]fimoduleOrecondition for all ### .."

E.PROOF. Let m

_{R.f}

and suppose that c is regular in Rt. Define ### &cR.fccRf

by### O(x)

cxfor all .v6### eR.f.

Clearlyis a mononaorphisn.

### Suppose

that a’6### eRr

and cx=0. Then_{c.rfRc}

0 which implies that
### xfRe

0 becausc c is regular. But then the left annihilator condition implies that .r 0 asrequired.

From Theorem 6and Proposition 5weknow that

_{eRf}

has finiteright uniform dimensionas
a right _{fRf-module.}

Since _{eRf}

and _{ceRf}

are isomorphic ### fRf-modules, d(ceRf)= d(eRf)and

so_{ceRf}

isanessential_{fRf}

submoduleof_{eR.f.}

Hence ### {y

fRf:y### ceR.f}

isanessentialrightideal of

_{fRf}

which, since _{fRf}

is semiprime right Goldie, must contain the required regular
element

_{c.}

IfS is semiprimeright Goldie, Q(S)denotes theclsicalright quotient ring of

### S

d if### M

is a right S-module,### Q(M)= M

sQ(S). Using the right common denominator property of Oresetswe see thatevery element ofQ(M) isofthe form m@

### c-

where M and cis regulinS.

### In

what follows weshallwriteTHEOM 3.2. If

### R

is a semiprime right Goldie generMizedmatrix ringwithidempotentsetE,then

### O(n)=

### O(nY).

e,

### f6

### E

PROOF. For each

### eE,

eRe embeds in Q(eRe) and we now check that for e,f### E,e

_{#}

f,### eRf

embeds in Q(eRf).### Suppose

that c is regular in fRf,### err

d zc=0.Then

_{fRezc}

0 and so _{fRez}

0 because c is regular in ### fRf.

Thus_{ezfRezf}

0 d hence
0

_{ezf}

zsince### R

issemiprime. Thisshowsthat_{eRr}

isatorsionfree_{fRf-module}

dso_{eRf}

embedsin

### Q(eRf).

Let e,f,9

### E,

### err,

### fR9

and suppose that c is regular in_{fRf}

and d is regul in
### 9R9.

Define### (ze-)(yd

### -)

### zycTd

### -

where Ya and### c

areobtned from the right bimodule Ore condition" yc eye. It is straightforward to check that this multiplication is well-defined and hat aresult### Q

### Q(eRf)

becomesageneralizedmatrixringwithidempotent setE.Wenowshow that s semiprime. Itfollows from Threm 6 that eRe is semiprimeright Goldie for all e

### E

and hence### Q(eRe)

is semiprime for all e### suces

toshow that### (Q, E)

satisfiesthe right nihilatorcondition. Let### yd-

### Q(fRe)

besuch that### (eRf)ud-’

0. Then### (eRf)(yd-’)

0andso### (eRf)u

0. From Corolly 1.2we sthat### (R,E)

satisfies the right annihilator condition and so y=0. Thus### (Q,E)

satisfies the right nihilatorconditionandhence### Q

is semiprime.Let e,

_{f}

### E,e

### f.

FromProposition 2.3 we seetha### err

hasfinite uniform dimension aright

### fRf-module

dso### Q(eRf)

has finite uniform dimensionas aright Q(fRf)-module. Since### Q(fRf)

is semisimple Artinian it follows that Q(eRf)isan artinian Q(fRf)-module, and hence### Q

is right Artinianbyan argument similarto### ([7], 1.1.7).

Since wehave already seen that### Q

issemiprime,

### Q

isasemisimpleArfinianring.To complete the prf, we need only show that

### R

isCLASSICAL QUOTIENT RINGS OF GENERALIZED MATRIX RINGS 315

bimodule Orecondition iffor eachmEMandeach regular elementcG5’thereisanm

### M

andaregularc

### T

suchthatme1### cm.

PROPOSITION 3.1. If

### R

is a semiprime right Goldie generalized matrix ring withidempotent set

### E,

then_{eRr}

satisfiesthe rightbimodule Oreconditionforall e,_{f}

E.
PROOF. Let rn

_{eRr}

and suppose that c is regular in ### ere.

Define### O:eRf--,ceRf

by### O(x)

cxfor all x### err.

Clearly0is an_{fRf-module}

homomorphism and wenowcheck that 0
is a monomorphism.

### Suppose

that x_{eRr}

and cx O. Then _{cxfRe}

0 which implies that
### xfRe

0 because c is regular. But then the left annihilator condition implies that x 0 asrequired.

FromTheorem6and Proposition 5weknowthat

_{eRr}

has finiterightuniform dimensionas
a right

### fRf-module.

Since_{eRr}

and_{ceRf}

areisomorphic ### fRf-modules,

### d(ceRf)= d(eRf)

andso

### ceRf

is anessential_{fRf}

submodule of### err.

Hence### {y

fRf:rny### e ceRf}

is anessentialright ideal of_{fRf}

which, since _{fRf}

is semiprime right Goldie, must contain the required regular
element

_{c.}

El
If 5’is semiprime rightGoldie,

### Q(S)

denotes the classicalrightquotient ring of5’and if### M

isa right 5,-module,

### Q(M)= M

(R)_{sQ(5").}

_{Using the right}common denominator property of

### Ore

setswe seethat every element of### Q(M)

is of the formm(R)### c-1

wherem### M

andcis regularin 5,.### In

what followsweshallwrite### rnc-

insteadofm(R)### c-1.

### THEOREM

3.2. If### R

is asemiprime right Goldiegeneralized matrixring withidempotent set### E,

then### Q(R)=

### y]

### Q(eRf).

e,_{f}

### EE

### PROOF.

For each### eE,

### eRe

embeds in### Q(eRe)

and we now check that for### e,f

### e E,e

### f,

### eRf

embeds in### Q(eRf). Suppose

that c is regular in### fRf,

x### err

and xc O. Then### fRexc

0 and so_{fRex}

0 because c is regular in _{fRf.}

Thus _{exfRexf}

0 and hence
0 _{exf}

xsince### R

issemiprime. This showsthat### err

is atorsionfree_{fRf-module}

andso### eRf

embedsin

### Q(eRf).

Let e,f,gE

### E,

x### _

eRf, y_{f}

Rgand suppose that cis regularin _{f}

### R

_{f}

and d is regularin
### gRg.

Define### (zc-1)(yd

### -x)

### xylcd

-1_{where}

yl and

### c

areobtained from the right bimoduleOre condition: ycl cyl. It is straightforward to check that thismultiplication is well-defined

and that as aresult

### Q

### Q(eRf)

becomesageneralizedmatrixringwithidempotent setE.### We

nowshow that### /.E

as semiprime. Itfollows from Theorem 6 that### ere

is semiprimeright Goldie for all eE### E

and hence### Q(eRe)

is semiprime for all e E. In view of Corollary 1.2, it suffices to show that### (Q,E)

satisfiesthe right annihilator condition. Let yd-### e

### Q(fRe)

be such that### Q(eRf)yd

-1_{0. Then}

_{(eRf)(yd-1)}

_{0}

_{and}

so

### (eRf)y

0. FromCorollary 1.2we seethat### (R,E)

satisfies the right annihilator condition and so y 0. Thus### (Q,E)

satisfies the right annihilatorconditionandhence### Q

issemiprime.Lete,

_{f}

### E,

e_{f.}

FromProposition 2.3 we seethat _{eRf}

hasfinite uniform dimensionas a
right_{fRf-module}

andso### Q(eRf)

hasfinite uniform dimensionas aright### Q(fRf)-module.

Since### Q(fRf)

is semisimple Artinian itfollows that### Q(eRf)is

anArtinian### Q(fRf)-module,

andhence### Q

is right Artinianbyan argument similar to### ([7], 1.1.7).

Since wehave alreadyseen that### Q

issemiprime,

### Q

isasemisimpleArtinianring.316 D.G. POOLE AND P. N. STEWART

property we can find, for each

### f

E### E,

an_{a!}

### _

fry and elements_{v(e,f)-eRf}

such that for all
### eE,x(e,f)=y(e,f)a[

### 1.

Let y=_{E}

y(e,f) and z= _{E}

a. Then x=Vz### -

and so### R

is aright orderin

### .

c,! E ! E!

E!

### Let R

be a semiprime right Goldie ring with idempotent e. Clearly,### R

is a generalizedmatrix ring with idempotent set

### E

### {e, 1-e}

and so it follows from Theorem 2.4 that### ere

issemiprime right Goldie. This result seems to have been well-known for some time. Also, it

follows from Theorem 3.2 that the classical right quotient ring of

### ere

is### eQe

where### Q

is theclassicalrightquotientringof

### R.

This resultisduetoSmall### [11].

### ACKNOWLEDGEMENT.

This paper was written while the first author was on sabbatical atDalhousie University. He would like to thank the

### Department

of Mathematics, Statistics and Computing Science forits warmhospitality.### An

earlier versionofthis paperincludedanunnecessaryfaithfulness assumptioninTheorem 2.4. The authorsaregratefulto MichaelV. Clasewho supplied them withapreprint which lead to the removal of that assumption.### REFERENCES

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