,
(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
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.
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
A
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)