Rings Whose Small Ideals Are Projective

，

(Department of Mathematics and Applied Mathematics，Huaihua University，Huaihua 418000，China)

Rings Whose Small Ideals Are Projective

XIANG Yue-ming*，WU Yi-qing

(Department of Mathematics and Applied Mathematics，Huaihua University，Huaihua 418000，China)

A right idealIof a ringRis small in case for every proper right idealKofR，K+I≠R.The ringRis called a rightJ-hereditary ring if every small right ideal is projective.RightJ-hereditary rings is developed as a generalization of right hereditary rings.Many properties of rightJ-hereditary rings are obtained.Several new characterizations of semiprimitive rings in terms of rightJ-hereditary rings are obtained.Related examples are given as well.

J-hereditary rings;small injective modules;covers

1 Introduction

Throughout this article，Ris an associative ring with identity and all modules are unitary.The Jacobson radical ofRis denoted byJ(R).Mn(R)stands for the full matrix overR.LetMandNbeR-modules.Extn(M，N)means ExtnR(M，N).For the usual notations we refer the reader to［1-11］.

A ringRis called right(resp.，left)hereditary if every right(resp.，left)ideal is projective.A right idealIof a ringRis small in case for every proper right idealKofR，K+I≠R.A rightR-moduleMis said to be small injective［5］if every homomorphism from a small right idealIofRtoMcan be extended fromRtoM.It is easy to verify thatMis right small injective if and only if Ext1(R/I，M)=0 for every small right idealIofR.The class of right small injective modules is closed under direct products and direct summands.In this paper，we say thatRis a rightJ-hereditary ring if every small right ideal is projective.We show that the concept of rightJ-hereditary is a proper generalization of right hereditary rings and semiprimitive rings(J(R)=0).Some interesting properties of rightJ-hereditary rings are obtained.

LetCbe the class ofR-modules.For anR-moduleM，a homomorphismg:C→MwithC∈Cis called aC-cover［2］ofMif the following hold:(1)For any homomorphismg':C'→MwithC'∈C，there exists a homomorphismf:C'→Cwithg'=gf.(2)Iffis an endomorphism ofCwithgf=g，thenfmust be an automorphism.If(1)holdsbut(2)may not，g:C→Mis called aC-precover.C-covers may not exist in general，but if they exist，they are unique up to isomorphism.We consider when every rightR-module has a monomorphic small injective cover.In light of this fact，we get a new characterization of a semiprimitive ring.

2 Main Results

Definition 1 A ringRis called rightJ-hereditary ring if every small right ideal is projective;equivalently，if every right ideal inJ(R)is projective.Similarly，we have the concept of leftJ-hereditary rings.

Remark 1(1)Obviously，every right hereditary ring is a rightJ-hereditary ring.

(2)Every semiprimitive ring is a right and leftJ-hereditary ring.

(3)Every rightJ-hereditary ring is a rightPSring(i.e.every minimal right ideal ofRis projective).In fact，in view of［4，Lemma 10.22］，every minimal right ideal ofRis either nilpotent or a direct summand ofR.

(4)It is easy to check that the direct product of any two rightJ-hereditary rings is also a rightJ-hereditary ring.

(5)SinceJ(Mn(R))=Mn(J(R))，J(eRe)=e(J(R))efor anye2=e∈Rand projective property is Morita invariant，being rightJ-hereditary is Morita invariant.

Example 1 LetR=K［x2，x3，yi，xyi］withKa field andx，y1，y2，…indeterminates overK(see ［6，Example 2.8］).ThenRis a non-coherent domain.LetGbe a free abelian group with rankG= ∞.Then the group ringRGis a non-hereditary semiprimitive ring.

It is well known that ifRis a right hereditary ring，then any submodule of a freeR-module is isomorphic to a direct sum of right ideals ofR(Kaplansky's Theorem).Similarly，we have the following result.

Theorem 1 LetRbe a rightJ-hereditary ring.Then any small submodule of a free rightR-module is isomorphic to a direct sum of small right ideals ofR.

The following is a characterization of a rightJ-hereditary ring.

Proposition1 The following are equivalent for a ringR:

(1)Ris a rightJ-hereditary ring.

(2)Every small submodule of a projective rightR-module is projective.

(3)Every quotient module of a small injective rightR-module is small injective.

(4)The sum of two small injective submodules of any rightR-module is small injective.

Proof(1)⇒(2)follows from Theorem 1.(2)⇒(1)is trivial.

Now，ifRis such that the intersection of two small injective submodules of any rightR-module is small injective，then the above result implies that any rightR-module is small injective，and henceRis semiprimitive.

Recall that a ringRis said to be semiperfect ifR/J(R)is semisimple and idempotents can be lifted moduloJ(R).Examples include right(or left)artinian rings and local rings.It is well known thatRis semiperfect if and only if every principal rightR-module has a projective cover(［7，Theorem B.21］).In the following proposition，we give a sufficient condition that rightJ-hereditary rings coincide with right hereditary rings.

Proposition3 IfRis a semiperfect ring，thenRis a right hereditary ring if and only ifRis a rightJ-hereditary ring.

Proof(⇒)follows by Remark 1(1).

(⇐).LetIbe a right ideal ofR.SinceRis semiperfect，R/Ihas a projective coverf:P→R/I→0.Then there is an exact sequence 0→K→P→R/I→0，whereK=Ker(f)is a small submodule ofP.By Proposition 1，Kis projective.Then，in view of Schanuel's Lemma，Iis projective.Therefore，Ris right hereditary.

Following［6］，a ringRis said to be a rightJ-semihereditary ring if any finitely generated right small ideal is projective.So a rightJ-hereditary ring is rightJ-semihereditary.By the analogous proof in［3，Example 2.33］，the ring in Example 2 is rightJ-hereditary and leftJ-semihereditary but not leftJ-hereditary.

Definition 2 A ringRis said to be rightJ-noetherian ifJ(R)is noetherian as rightR-module.Similarly，we can define leftJ-noetherian rings.

Remark 2(1)Right noetherian rings are rightJ-noetherian.

(2)Every semiprimitive ring is right and leftJ-noetherian.

(3)By［8，Theorem 2.17］，Ris rightJ-noetherian if and only if every direct sum of small injective rightR-modules is small injective.

Proposition4 If every rightR-module has a small injective cover，thenRis rightJ-noetherian.

Proposition5 The following are equivalent for a ringR:

(1)Ris a rightJ-noetherian and rightJ-hereditary ring.

(2)Every rightR-module has a monomorphic small injective cover.

ProofIt follows from Proposition 1，Proposition 4 and［9，Proposition 4］.

Remark 3LetRbe a local algebra of finite dimensionn＞1 over a fieldF，and the unique maximal idealT≠0.By the argument of［3，p.51］，Ris right artinian but not right hereditary.SoRis rightJ-noetherian but not rightJ-hereditary by Proposition 3.Then there exists a ring whose every rightR-module has a small injective cover but need not be a monomorphism.

Corollary 1A ringRis semiprimitive if and only ifRis a rightJ-noetherian，right small injective and rightJ-hereditary ring.

Proof(⇒).It is clearly.

(⇐).By Proposition 5，every rightR-moduleMhas a monomorphic small injective coverf:E→M.SinceR

is small injective as a rightR-module，the homomorphismfis epimorphic，soMis small injective.By［5，Proposition 3.3］，Ris semiprimitive.

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O153.3

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1000-2537(2012)01-0001-04

Small理想都是投射的环

向跃明*，吴毅清

(怀化学院数学与应用数学系，中国怀化 418000)

若对任意真理想K，有K+I≠R，则称环R的右理想I为small理想.若任意small右理想是投射的，则称环R为右J-遗传环.引入右J-遗传环作为右遗传环的推广，给出了右J-遗传环的一些例子和性质.利用右J-遗传环得到了半本原环的一些新刻画.

J-遗传环;small内射模;覆盖

2011-03-19

国家自然科学基金资助项目(11171102)

*

，E-mail:xymls999@126.com

(编辑 沈小玲)