Pure Baer injective modules

Internat. J. Math. & Math. Sci. VOL. 20 NO. 3 (1997) 529-538

529

PURE

BAER

INJECTIVE MODULES

NADA M. AL THANI DepartmentofMathematics

Faculty ofScience QatarUniversity

Doha,QATAR

(ReceivedApril 19,1995and in revisedform April 16, 1996)

ABSTRACT. Inthispaperwegeneralizethe notionof pure injectivity of modules by introducingwhat wecall apure Baer injectivemodule. Somepropertiesandsomecharacterizationof such modulesare established. Wealso introduce two notionscloselyrelatedtopure Baer injectivity;namely,the notionsof

a

E-pure

Baerinjective module and that of SSBI-ring.

A

ring

R

is anSSBI-ring ifandonlyifevery smisimple R-moduleispureBaerinjective. Toinvestigate such algebraicstructureswehad to define what we call p-essential extensionmodules, purerelativecomplement submodules,leftpure hereditary tingsand someother relatednotions. Thebasicproperties of theseconcepts and theirinterrelationships

areexplored,andarefurther relatedto thenotionsofpuresplit modules.

KEYWORDS AND PHRASES: Pureand

E-pure

Baerinjectivemodules, pure hereditary ring, pure-splitmodule,P-essential extensionsubmodule,purerelativecomplementsubmodule, SSBI-ring.

1991AMSSUBJECT CLASSnZICATION CODES: 13C.

0. INTRODUCTION

Inthis introduction we establishterminology and recall a summary ofsome basicdefinitions and

resultsintheliteraturenecessaryforsubsequentsectionsofthepaper.

Throughout

R

willdenotearingwithidentity. Unlessotherwisestated, allmodules willbe unitary leftR-modules,and allhomomorphismswillbe R-homomorphisms.

A short exact sequence 0---, N---} M---} K--}0 of left R-modules is said to be pure if" L(R)

RN

L(R)

RM

is amonomorphism forevery right R-moduleL. AsubmoduleNofanR-module

Miscalledapuresubmoduleof

M

incasethenaturalhomomorphismL(R)

sN

L(R)

M

isinjective

for every right R-module L. Equivalently, N is pure in M if and only ifforany finite system of linear equations

r,z

a,,1

<

<

m, _{where %}E

R

and a, E

N,

ifthe system has a solution

3=1

(sl,

sn)

M

n,

italso hasasolution

(tl,

t,)

N".

Aleft R-moduleisregularifand onlyifevery submodule ofMispure,and asubmoduleNofaflat R-moduleMispureifIN

IM

NNfor all rightidealsIofR. Forfurther resultsconcerningthis type

of puritysee[1 and[2].

Aleft idealI ofaring

R

ispurein

R

if andonlyiffor everyz Ithere exists anV Isuch that z zV Furthermore,aring

R

isVon-Neumannregularifand onlyifeachleft(fight) R-moduleis flat if andonlyifevery (principal) left (fight)ideal ispure

(see

[3]).

530 NM AL THANI

idealisprojective. Formodulesover aPrefer ring, purity and RD-purity coincide, flatness andtorsion ffeeness coincide(see[4]).

By K MweshallunderstandthatKis an essential submodule of

Aring

R

iscalledaleft V-ringif andonlyifevery simple left R-moduleisinjective. Kaplansky

provedthat a commutativering

R

is aV-ringifandonlyif it isregular[5].

Aleft R-moduleM iscalledpureinjectiveif it isinjectiverelative toeverypureexactsequence of R-modules Warfield

[4]

provedthatany left R-modulecanbe embeddedas apure submodule ofapure

injective R-module. Itcanalso be proved thatapureinjective R-moduleisa directsummandofevery

R-modulecontainingitas apure submodule(see

[6]).

For further relatedresults wereferto[2],[3]and[4],togetherwiththemonographs ], [6]and[7]

1. PUREBAER INJECTIVE MODULES

Wenow introduce thedefinitionofapureBaerinjective module

DEFINITION 1.1.

An

R-module

M

iscalled apure Baerinjective module if for eachpureleft

ideal

I

of

R,

any R-hemomorphism

f

I

-

M

canbe extendedtoanR-homomorphism

f

R

Ifaring

R

is freefromnon-zero onesided zerodivisors thenany R-moduleisnecessarilypure Baer

injective. Infact

R

doesnotpossess anynon-zeroproper pureone-sidedideal in this case. Thismeans,

inparticular,thatanyabeliangroupis apureBaerinjective Z-module.

Wefind itnecessarytopoint outfrom thestartthat the notionofpure Baerinjectivityis different

from that of pure injectivity; asanexampleconsiderthe Z-moduleZ. However,it isevidentthat every

pureinjective R-moduleispure Baerinjective. Furthermore, we notethe easily deduced fact that any pureBaerinjective module overaVon-Neumarm regularring is

injective.?

The following result is essential in characterizing pure semisimplicity ofrings, a notion to be introduced in thesequel.

THEOREM1.2(Pure BaerInjectivityTest). Foraleft R-modulethefollowingareequivalent

(1) MispureBaerinjective

R-module

(2) Forevery pure left idealIofRandevery R-homomorphism

f"

I

M,

there exists an m EM

such that foralla

_

I,

f

(a)

am;

(3)*

For every pureexactsequence

thesequence

0

I

R

R

/

I---,0

M

Homl(I,M)

0

wish tothank my study supervisors forcallingmyattentmntothisresult. isexact.

PROOF. Clear. VI

PROPOSmON1.3. The directproduct 1-I,M, ofR-modules ispure Baer injectiveifandonlyif

each

M,

ispureBaerinjective.

PROOF. Clear.

Werecall that a moduleMover aring

R

iscalledtorsion-free ifforno0:/:r R,rz 0 unless O=xq.M.

Theproofof the previous result showsthatif However,wehave:

PROPOSITION1.4. Adirect sum

oMo

oftorsion-freeR-modulesispureBaerinjectiveifand onlyifeach

Mo

ispure Baer injective.

PROOF. Let

Mo

be pure Baer injective for each c and consider any R-homomorphism

PUREBAER INJECTIVEMODULES 531

applying the pure Baer injectivity test Lemma 1.2, one can find

rno

E

Mo

for each a such that

raf(x)

Xmo for everyx EI. Thus

f(x)

_,

Xmo. Thismeansthat

xrna

0forall but a finite

fimtc

number of the indices. Butsince each

Mo

istorsion-free,weconclude that

mo=

O for almost all

This meansthat theelement

{too}

of

HoMo

isanelement of

oMo,

proving thatthis direct sum is

pureBaerinjective I"1

2. THE NOTIONS OF PURE HEREDITARY AND PURE SIMPLE RINGSANDTHEIR ROLEIN PUREBAER INJECTIVE MODULES

Inthis section we define andstudytwonotions: thepurehereditary ringand thepuresimple ring, both ofwhichappeartohaveavitalrole incharacterizingpureBaermodules.

DEFINITION2.1. Aring

R

iscalled leftpurehereditaryifeverypureleftidealof

R

isprojective.

Inwhatfollowsweshall provethat theclass of pureBaerinjective R-modules overapure hereditary ring,ishomomorphicallyclosed.

THEOREM2.2. Thefollowingstatements arepair-wise equivalent foragiven ring

R

(1) R

isleftpurehereditary;

(2)

The homomorphic image ofapure Baerinjective R-moduleispure Baerinjective;

(3)

Thehomomorphic image ofaninjective R-moduleispureBaerinjective;

(4) Anyfinitesum of injective submodules ofanR-moduleispure Baer injective.

PROOF. (1)

=

(2)

Considerthe following diagram

0 I R

M

-g

K

0

ofR-modules,where

I

is apure leftidealof

R,

and

M

ispure Baer injective. Projectivity of

I

shows

that for some R-homomorphism

"

I--,M,

f

go. Moreover there exists a homomorphism

"

R

M that extends

.

Thisshows that

f

g

is anextension off; and so

K

ispure Baer

injectiveasrequired.

(2)

=,(3) Clear.

(3)

=

(1)

Let Ibeapure leftidealof

R

andMbe aleftR-module whose injective hullis

E(M)

Consider the following diagram of R-homomorphisms

E(M)

g-

K 0

recallingthat

K

ispure Baerinjective by assumption. So,there exists anR-homomorphismh

R

K whose restriction on I is

f.

Again since

R

is projective, there exists an R-homomorphism

a:R--,

E(M)

suchthatga h; andso gai

f.

This means thatIisE(M)-projective ThusIis M-projective

(for

theproofcf [5, p. 180,prop 16.12]);andso

I

isprojective

(4)

=

(3) Let Nbe asubmodule ofaninjectiveR-moduleE. Toprovethat

E/N

ispureBaer

532 N M AL THANI

M1

{(x,

O)

4-K x

_

E}

and

M2

{

(0,

y)-+-

K

y

_

E}

ofQ/K. Now Q/K

M1

4-M2,also,

MlnM2-{(x,0)+K-xeN}.

For ifxeN, then (x,0)+K--(0,-x)+KeMlf’IMg. on theother hand theassumptionthat

(x, 0)

4-K

(0,

y)

+

Kwherex,y EEmeansthatx y

e

N

and therefore

M1

fq

M2

{

(x, 0)

+

K"x E

N}

Define

f-

E

--

M1

with

f(x)

(x, O)

+

K

forall

xE

E

to obtain the isomorphisms E

=" M1

and N

M1

f’l

M2.

Similarly E

M2.

Thus

Q/K

M1

+

M2

ispureBaerinjective. ThisshowsthatforsomeR-module

G,

Q/K

M1

$Gand G

(Q/K)/M1

M2/M1

f’l

M2

E/N.

Now since G is pureBaerinjective,

E/N

ispureBaer

injectiveas tobe proved. IZ!

If

R

isleft self-injective, the pureBaerinjectivity of each homomorphic image

RR

canbediscussed in viewof thefollowing.

PROPOSITION2.3. Let

R

be aleft self-injective ring. If

R/J

ispureBaerinjective foreach essentialleftideal

J,

then

R/I

ispureBaerinjective forevery leftidealIofR.

PROOF. Let

I

be a left ideal of

R

and let

E(I)

be its injective hull.

Now,

since

E(I)

is a direct summand of

R,

there exists an idempotent e

E(I)

such that

E(I)

Re.

Consider the

R-homomorphism

f-

R

---,_re,with

f(r)

refor eachr

e

R. Since

I

Re,

f-1

(I)

R. Therefore by hypothesis

R/f-I(I)

ispure Baer injective. Define

]"

Re/I

R/f-l(I)

by

](re +

I)

r

+

f-l(I)

This is awell-defined R-isomorphism; and so

Re/I

ispure Baer injective. We provethat the sura

B

R(1

e)

+

I

is a direct sum. Ifx

R(1

e)

f’lI, thenx r re

r’e

forsome r,

r’

.

R

and so x 0. Thus

B/I

"

R(1- e)

showing thepureBaer injectivity of

B/I.

Furthermore wehave

R/I

B/I

Re/I.

To see thiswe notice that r

+

I

(r(1

e)

+

I)

+

(re

+

I)

for each rE

R

Also 5

r(1

e)

+

I se

+

I

means that r re

(s

+

r’)e

for some

r’

R

which shows that

5 0. Now

R/I,

being thedirect sum oftwo pureBaerinjective R-modules, should be pureBaer

injective.

IZI

DEFINITION2.4 [$]. AleftR-module

M

iscalled pure-splitifevery puresubmoduleofMisa directsummand.

R

islett pure-splitif

RR

ispure-split.

Itobviously follows that every left pure-split ringisleft pure hereditary.

Itiseasilyseen thateverypuresubmoduleofapure-split moduleispure-split,and thequotient ofa

pure-split module byapure submoduleisagain pure-split.

Wethusextractthe following simple result

TItEOREM2.5. Foragiven ring

R

the followingstatementsareequivalent.

(1) Everydirect sumof copies ofRispure-split;

(2) Everyflat R-moduleispure-split.

PROOF. (1)

=

(2)

For anyflat module

M,

there is apureexactsequence

O---K--, F--,

M

_O

whereFisfreeand so is apure-split module; consequently bothKandMarepure-splitmodules. (2)

=

(1) Clear. E!

TItEOREM2.6. Thefollowingstatements arepair-wise equivalent foragiven ring

R

(1)

R

isleftpure-split;

(2) EveryleftR-moduleispure Baer injective;

(3) Everypureleft ideal ofRispure Baer injective;

(4) Everypure leftidealof

R

isprincipal.

PROOF. (1)

=

(2) LetIbe apureleftidealof

R

andlet

f

I MbeanR-homomorphism

Then

R

I JforsomeleftidealJof

R,

so that there is anR-homomorphism

R

Mthat extends

f

(2)

=

(3) Obvious.

(3)

=

(1)

Sinceevery pure leftidealof

R

ispure Baer injective, each pureexactsequence

PUREBAERINJECTIVEMODULES 5 3 3

splits, showingthat

R

isleft pure-split.

(4)

:=

(1) LetMbe anR-moduleand letI

Rz

be apure leftidealofR. So,forsomeaEI, x za. If

f"

Rz

M is agiven R-homomorphism, therestrictionon

I

of the R-homomorphism

f"

R

M

effectedby

f(1)

f(a)

is

f.

Indeed

f(rx)

f(rza)

f(rx)

forallr

(1)

=

(4) Obvious.

REMARK2.7. We knowthat

RR

issemisimpleifandonly if everyR-module issemisimple. This

fact cannot be extended to pure-splitting To see this we notice that Z is a pure-split Z-module

However 7"‘ canbe embedded as apure submodulein apure injeetive Z-module o

r.

But o cannotbe

pure-split since otherwise 7’, _{will be a directsummand ofo}

r,

_{contradicting the fact that}

Z

_{is not pure}

injective. El

However,

the previous propertyis validforcertaintypes of rings. For example, aregular ring

R

is

pure-splitifand onlyif it issemisimple. Wecanthusstatethefollowing proposition.

PROPOSITION2.8. Foraregular ring

R,

thefollowingstatements areequivalent:

(1)

R

ispure -split;

(2)

Every

moduleissemisimple;

(3) Everymoduleispure-split;

(4) Every(pure)exactsequenceissplitexact.

PROOF. Clear. El

Wenowrecallthefollowing definition; for analogousandrelatedconceptcf.[6,p.48].

DEFINITION2.9. AnR-moduleMis calledpure injectiverelativetotheR-module

N,

orsimply

N-pure

injective,if ineachdiagram

0

K

-f

N

M

wherethe embedding

f(K)

is pure in

N,

there exists an R-homomorphism h-N M such that

g=hf.

Thefollowingtheorem relates some of thepreviousnotions.

TItEOREM2.10. ForaleftperfectringRthefollowing properties hold:

(1) EveryfiatR-moduleispure-split,

(2) EveryR-moduleispure Baer injective, and

(3) EveryR-module ispure injectiverelative toanyfiatR-module.

PROOF. (1)and (2)arebothdirect. Toprove(3),letCbe a lettR-module andNbe apure submodule ofa fiat R-moduleM. ThenN isa directsummand of

M,

so that every homomorphism

N Cextendstoahomomorphism

M

--,_C. l’i

DEFINITION2.11(see

[9]).

Anon-zeroR-moduleMiscalled pure simpleif

(0)

and

M

are its

only pure submodules.

PROPOSITION2.12. Inacommutative ring

R

in whicheveryideal isthe intersectionofmaximal

pure ideals, every epimorphism

f"

I S whosedomainis apureidealof

R

andcodomain is a pure simple R-modulehas an extension

f

R

S.

PROOF. Let

f"

I

Sbeasgiveninthe premise

By

assumptionKer

f

isthe intersection ofa

family

(C:

j

J)

ofmaximalpureidealsofR. IfI

c_

C’

for all j

J,

then

Kerf

Iand this means thatS

(0)

contradictingthatS is apure simple R-module. Thus forsome j Jwehave

I

C

534 N M ALTHANI

pure in R. SoICIC is pure inI, and so

I[3Ca/Kerf

is a pure submodule of

I/KerfS

By

assumptioneitherI C

a/Ker f

{ 0},

orI[3C

a/Ker f

I/Ker

f.

The latter assumptionisimpossible,

since otherwiseICC_a. So, INC Ker

f.

Nowtheassignment

f-

R Sdefinedby

f(r)

f(i)

where EIthat satisfiesr

+

c forsome cEC is a well-defined functionthat gives the required homomorphism.

COROLLARY2.13. Let

R

bea commutative ring in whichevery ideal is the intersection of

maximalpureideals. Then

(1) Every

pure simple R-moduleispureBaerinjective,

(2)

Every

semisimple R-moduleis a direct sumof pure Baer injective R-modules.

PROOF. The proof followsfromProposition212.

LEMMA2.14. Afinitelygeneratednon-zeroR-moduleMpossessesamaximalpure submodule

PROOF. The set

A

(K-

K

<

MandKpurein

M}

is partially ordered by inclusion. The

unionofachainof

A

isclearlyamember of

A;

and anappealtoZom’s Lemmayields the result.

THEOREM2.15. Let

M

beanR-module

M

in whichevery cyclic submoduleispure-split,then

every non-zero submodule ofMcontains apure simple submodule.

PROOF. Let

(0}

-

K

_< M,

if 0 x EK. Then byLema2.14

Rx

contains a maximal pure submodule, say,H. ThusRx H H

.

Thisshows thatH is anon-zero puresimplesubmodulein

Rx

andso isa non-zeropure simple submoduleinK.

Itseemsthatanappropriate notion ofan"intersection

property"

would playanimportantrole in the structuretheoryof rings. Thuswithin ourcontext,wedefine

DEFINITION 2.16. An R-module M is said to have the pure intersection (resp. pure finite intersection)propertyifand onlyiftheintersectionof any(resp.finite)family of pure submodules ofM isagainpure.

Itcanbe easily shownthatanycommutativeringpossessesthepurefinite intersection property, and furthermore,aregular ring

R

possessesthepureintersectionpropertytoeach R-module

PROPOSITION 2.17. Any torsion-free R-module

M

over a Pfer ring

R

has the pure

intersectionproperty.

PROOF. Itis known that over a Pfer ring purity and RD-purityare equivalent notions, see [4,p. 706]. Thus therequired resultis immediate if we noticethatthe intersectionof any family of

RD-submodules ofatorsion-freemodule isagainanRD-submodule,see 10, p.39].

PROPOSITION2.18. Atorsion-freeR-moduleMoveraprincipal rightidealringRhas thepure intersection property.

PROOF. Let

{

N j E

J}

beafamily ofpuresubmodules ofM. Inviewofthetorsion-freeness of

M, K

fjNis anRD-submodule, see[10,p.

39].

We provefirstthat

K

is an f-puresubmodule.

Tothis endletIbeagiven rightidealofR. NowK is

RD-pure

and

R

is aprincipal rightidealring

So,givenx K

IM,

we can finda Iandm Msuchthat x am aK. ThusK IM IK,

showing that

K

is -pure. Finally the flatness ofMguaranteesthepurity ofK.

3. THE,BI-RINGAND -PURE BAERIN$ECTIVEMODULE

Inthis section we introducetwo related notionsnamely SSBI-ring and

T-pure

injectivemodule whichprovetobeusefultoourinvestigations.

Byrd

I11

callsaringRanSSI-ringifevery semisimple R-moduleisinjective. Inwhatfollowswe

generalizethisconcept.

DEFINITION 3.1. Aring

R

is called anSSBI-ringifevery semisimpleR-module ispure Baer injective.

PUREBAERINJECTIVE MODULES 535

PROOF. Let

I0

C

I1

C C

Im

C be astrictly ascendingchainof pure leftidealsof

R

and for a

given

Ik

inthechaintakea

Ik

such thata

Ik-1.

Thefamily

Ak

L <_

R

Ik-1

<_

L

< Ik

a

L}

ispartiallyorderedbyinclusion in whicheverychainhas anupperbound. LetLkbeamaximalmember

of

Ak.

Thus, Ra

+

Lk/Lk

is simple and the given assumption yields

Ik/Lk

Ra

+

Lk/Lk

N/Lk

forsomeleftidealNofR. Now,sincea

N,

the maximality of

Lk

forcesN

Lk.

This meansthat

Ik/Lk

is simple. Hence

l/Lk

lk/Lk

(

Nk/Lk

for some left ideal

Nk

and I U

Ik. Now,

if

K

"

x

+

Lk

"k

+

_kforsome

k

Ik/Lk

andk

Nk/Lk,

theassignmentcr I

(klk/Lk,

with

c(x)

{k

}

isawell-definedR-homomorphism. Forif z

I,

then x

L

forsome anda:

L+3

for

all j.

So,

or(z)=

{k}

(k(lk/Lk)

since

+3

0and

(k(Ik/Lk)

issemisimple.

By

hypothesis

k(Ik/Lk)

is therefore pure Baer injective and we should have the extension R-homomorphism

-"

12-’*

(k(lk/Lk)

of tT. Furthermore, since

(1)6-

’=I(I,/L:)

for some r Z

+,

we see

that

(I)C_

,=I(I,/L,).

Suppose now that =z+l+,

I,+/L,+,

and

or(z)= (k),

then

,+, 0. The argument shows that

I,+

L,+,,

contradictingthefact

L.+

A+z.

Thisshows

thatthe abovetowerof pureleft ideals isoffinitelength. I"!

DEFINITION3.3. Aleft R-module

M

is called

E-pure

Baerinjectiveifeverydirect sumof copies of

M

ispureBaerinjective.

As examples of

E-pure

Baer injective modules we mentiontorsion-free modules, modules over

integraldomains orover left pure-split rings.

THEOREM 3.4. Aring

R

in whichevery injective moduleis

E-pure

Baer injective, satisfiesthe ascendingchaincondition onpure leftideals.

PROOF. Let

I1

C

I2

C C

In

C be achainof pure left ideals Foreach let

K,

be the

injective hull of

R/I,

and let K

,K,.

For every c

Z

+,

YI,K,

Ko

II,oK.

Ifwe set

Mo

IIK,,

then

Mo

isinjective Byabuse ofnotation wehave

By assumption

B,Mo

is pure Baer injective. Thus K itself is pure Baer injective. Now the

R-homomorphism

f"

OiI,.-.*

K,

defined by

f(z)=(z+Ii}

extends to an R-homomorphism

"

R

---,_{K. Letn}

Z

+ _{such that}

(1)

(3=lKi.

Then

f(t3I,) <

IK.

So,if z O,Ithen x

Io

for alla

>

n, andso

PROPOSITION3.5. Adirectsummand ofa

E-pure

Baer injective moduleisagain

E-pure

Baer

injective.

PROOF. Immediate.

THEOREM 3.6.

R

is left pure-split ifand only ifR is left pure hereditary and

E-pure

Baer injective.

PROOF. e= Let

I

beapureleftidealof

R.

Then

I

isprojective andso adirectsummand ofafree R-moduleF. But

R

ispureBaerinjective. Thus bothFand

I

arepure Baerinjective, yieldingtheleft pure-splitting ofR.

:

Theprooffollows fromTheorem 2.6. El

4. P-ESSENTIAL SUBMODULES

Inthis section we introduce the notionof p-essential submodules withthe aimof gaining further insight about the structure of pure Baer injective modules. For example, Corollary 4.16 to follow presentsacriterionforaleft R-moduleover acommutativering

R

tobepure Baerinjective, employing

thisnotion ofiv-essentiality. Alsobyusingthisnotion, corollary417sharpensaresult[5]ofKaplansky’s onV-tings.

DEFINITION4.1. AsubmoduleKofanR-moduleMis calledp-essentialin

M,

abbreviatedby

536 NM.ALTHANI

Miscalled a p-essential extensionofK. Also amonomorphism

f

N

M

iscalleda p-essential if Irn

f

s’M.

Given left R-/nodules

M

and N,

PHa(M,N)

designates the set of R-ho/no/norphis/ns

PHR(M,N)=

{h"

h"M NandKerh purein

M}.

PROPOSITION 4.2. The following state/nents are equivalent for any sub/nodule Kofan

R-/nodule

M

(1)

K

(2) Foreach R-moduleNandfor each hE

PHIt(M, N)

(Ker h)

fl

K

0}

impliesthathismonomorphis/n

PROOF. (1)

=

(2)

Holds bythe definitionofp-essentialsub/nodule.

(2)

=

(1) Sinceany puresub/nodule

L

ofMisthe kernelforsomeh

.

PHIt(M,A)

forsome

R-module

A,

then

L

gl

K

{0}

_{implies that}

L

{0}

andso

K

s’M.

V1

COROLLARY4.3. A/nono/norphism

f

K

-

M

is p-essentialifandonlyif(anyepi/norphism) h

e

PH/i

(M,

is/nonicwhenever

hf

is/nonic

PROOF. Let0 K

-/M

bep-essentialandh M

N,

where h

.

PHIt(M,

N).

Thenif

hf

is monicKerhflIm

f

{0}.

So,Kerh

{ 0}

and h is monic. Conversely, supposethat

af

K M is a /nono/norphis/n satisfying the given condition and let

L

be a pure sub/nodule of M with

L

flIrn

f

{0}.

Then

rf

is obviously /nonic, r being the canonical /nap zr"

M

M/L.

By

assumptionrshouldbe/nonic. This/neansthat

L

{0}

andIrnfs’M.

Thefollowing resultisanalogousto a similarresult concerningessential sub/nodulesofamodule.

TREOREM4.4. LetK

<

N

<

Mbeatowerof R-modules. Then: (1) IfK

s’M

then

NsrM.

(2) IfNispureinMandK

rM,

thenK

sN

andN

(3) IfM has purefinite intersectionpropertyand ifNispurein

M,

then

K

s’M

ifandonlyif K

s’N

andN

PROOF.

(1)

and

(2)

are obvious.

(3) Let

L

be purein

M

with

L

N

K

0.

Byassumption

L

NNispureinM. Thismeansthat

L

NNispureinN. ThusL N

₀

and consequently

L

0.

ThereforeKS

COROLLARY4.5. LetMbeanR-module thathas thepurefinite intersectionproperty. IfHis

pureinM,thenHNK

_

’M

ifandonlyifHS

M

and

K

S

rM

for anysubmodule

K

ofM.

PROOF.

::

The proof follows fromTheorem 4.4. Supposethat

H

s’M

and

K

’M.

Given apure submodule

L

of

M

with

L

(H

N

K)

(0,

then

L

H

is

pue

in M byhypothesis. Thus

L

H

{0}

andconsequently

L

{0}.

l-!

EXAMPLE

4.6. Wegive hereanexampleofsub/nodules

A, B, A’

and

B’

ofa certainZ-module

M Z

Z/2Z

with

AS

’B

and

A’

s’B

whereasA

+

A’

isnotp-essential inB

+

B’.

The idea ofthis counterexampleis lifted fro/nexample 1.2from Goodearl’s/nonograph

[7].

Take

A

A’

Z(2,

),

the sub/nodule generated by

(2,),B=Z(1,)

and

B’=

Z(1,).

Now

AZ(0,)=

{0}

see

Goodearl[7]. Whatisleftnowtoproveour assertion istoprove that

Z(0, )

ispureinB

+

B’

Tosee thissupposethat

n(m,

)

+

(k, k--)

(0,

).

This meansthat

L

isodd andm k. Againnisodd

This gives

n(m,

"6)

+

(k, 1-)

n(O,

),

showing that

Z(0,

$)

ispurein

B

+

B’.

Apure leftideal

I

ofaring

R

is a directsummandof

R

if

I

ispure Baerinjective.

Hence,

wehave

thefollowing:

PROPOSITION4.7. Aring

R

cannot have aproper pure leftidealIwhich isbothp-essentialand

pure Baer injective.

PURE BAER INJECTIVE MODULES 5 3 7

PROPOSITION4.8. Apure injective moduleiVdoesnothaveaproperp-essential extension

M

in whichNispure.

PROOF. Clear. I-I

COROLLARY4.9. AnyR-module

M

cannothaveapropersubmodule whichisboth injectiveand p-essential.

PROOF. Anyinjective submodule

K

ofMis pureinM. But

K

ispure injective. So,if

K

is p-essential in

M,

theprevious proposition showsthatK M.

DEFINITION4.10. AsubmoduleNofanR-module

M

iscalledapureclosed submodule of

M

if

M

doesnotcontainaproperp-essential extensionofN. ObviouslyN

_<

M

ispure closed if andonlyif

N__<’K < M

implies that

K

N.

PROPOSITION4.11.

Any

directsummandofanR-module

M

is pure closed.

PROOF.

LetM=AB.

If

A

<_’K

<_ M,

thenKfBispureinK

ButKnBqA={0}

Thismeatusthat

K

N

B

{0};

andso

K

A.

COROLLARY4.12.

(1) Every

pureinjective R-module

M

ispure closedinanyR-modulethat contains

M

as apuresubmodule.

(2)

Apure leftidealof

R

which ispureBaer injectiveispure closed in

PROOF. (l) LetMbeembeddedinNas apure submodule. Inthis caseMisa directsummand

of

N;

and so

M

should bepureclosedin

M

(2)

Clear.

DEFINITION4.13. Let Nand

K

besubmodules ofanR-moduleMwith

K

purein

M

K

is

called purerelativecomplements ofNin

M

ifKismaximal with the property

K

f3N

0}.

PROPOSITION4.14. EveryR-submodule of

M

hasapurerelativecomplementinM. PROOF. Let Nbeagiven submodule ofMandconsider theset

A

{K;K

< M,K

purein

M

andN

nK

{0}}.

A

is partially ordered byinclusion. Obviouslyanychainof

A

has an upper bound. Zorn’slemma then

guaranteesthat

A

has a maximalmember,whichmeansthatNhasapurerelativecomplementinM IZi PROPOSITION4.15. I and Jare givenidealsofacommutativering

R

IfJ ispurerelative

complements ofIin

R,

then

I

Jisp-essentialinR.

PROOF. Let

A

N

(I

J)

{0}

for some pureideal

A

of

R

ThenI

n (A

J)

{0}

Since

R

iscommutative,

A

JispureinR. The maximality ofJforces

A

{0}.

This means that

I

9Jis

p-essentialinR.

Sowededuce thatacommutativeringthathasnoproperp-essentialideals’isnecessarily semisimple.

COROLLARY4.16.

R

is acommutative ring. AnR-moduleMispure Baer injectiveifand only

ifHomR(R,

M)

HomR(J, M)

is anepimorphismforeverypure,p-essential idealJofR.

PROOF. Consider ahomomorphism

f

I MwhereIis apureidealof

R

ByProposition4.14 we can findapurerelativecomplement Jof

I

inR. Then

f

extendstoa homomorphism

I

9J

M,

andthisextendstoahomomorphism

R

M

byourassumption.

Itisknown

[5]

that if

R

is aright V-ring,then/ Ifor any rightidealIofR. Thisyieldsthe celebratedresult ofKaplanskystatingthat a commutativeringis aV-ringifandonlyif it isregular The

followingresultrefinesthatofKaplansky’s.

COROLLARY4.17. AcommutativeringRis aV-ringifandonlyifI

12

for every p-essential

ideal

I

ofR.

PROOF. Let Jbeanideal of

R.

If

J’

is apurerelativecomplementofJin

R,

thenJ

+

J’

is

538 N M.ALTHANI

ACKNOWLEDGMENT. This work comprises a portion of a Ph.D. thesis written under the supervision of ProfessorIsmailA. Amin, ProfessorJavedAhsanandProfessor Mohammed Seoud. The author wishes to thank them all for suggesting this problem, and also for some helpful comments

throughout thedevelopmentofthis work.

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[3]

FIELDHOUSE, D.J.,Regular tingsandmodules, J.Austral. Math.

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WAKFIELD,

R.B.,Jr.,Purityandalgebraic compactness for modules,

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J.Math.,28(1969),

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ANDERSON,

F.W. and

FULLER, K.R.,

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