On Strongly Regular Rings

BAN Xiu-he,LIANG Shi-jian,YIN Chuang

(1.School of Mathematical Sciences,Guangxi Teacher’s Education University,Nanning 530001,China; 2.Nanning Number 18 Middle School,Nanning 530001,China;3.Guangxi Academy of Sciences,Nanning 530007,China)

On Strongly Regular Rings

BAN Xiu-he1,LIANG Shi-jian2,YIN Chuang3

(1.School of Mathematical Sciences,Guangxi Teacher’s Education University,Nanning 530001,China; 2.Nanning Number 18 Middle School,Nanning 530001,China;3.Guangxi Academy of Sciences,Nanning 530007,China)

In this paper we investigate strongly regular rings.In terms of W-ideals of rings some characterizations of strongly regular rings are given.

strongly regular ring;W-ideal;p-injective module

§1.Introduction

All rings in this article are associative rings with identity,all modules are unital left modules unless otherwise stated.Recall that R is(von Neumann)regular if for every a∈R,there exists some b∈R such that a=aba.R is strongly regular if for every a∈R,there exists some b∈R such that a=a2b.Strongly regular rings is left-right symmetric.A ring is a reduced ring if it contains no non-zero nilpotent elements.Strongly regular rings are regular ones and a ring R is strongly regular if and only if R is a reduced regular.A ring R is a duo ring if every left (right)ideal of R is a two sided ideal.A module M is called p-injective if for any principal left ideal I of R and any left R-homomorphism g:I→M,there exists y∈M such that g(b)=by for all b in I.A ring R is said to be left p-injective ifRR is p-injective.

Following Haiyan Zhou[1],a left ideal L of a ring R is called a weak ideal(W-ideal),if for any a(/=0)∈L,there exists n＞0 such that an/=0 and anR?L.The ideals of R are W-ideals ofR and the converse is not true.In this paper,we shall give some characterizations of strongly regular rings in term of W-ideals of rings and p-injectivity,generalizing some known results.

Throughout this paper,the notations A≤B and C≤eD will mean that A is a submodule of B,C is a essential submodule of D,respectively.We will write l(a)to indicate the left annihilator of an element a of a ring R.

§2.Main Results

We begin with two lemmas.

Lemma 2.1[2]Regular rings are nonsingular rings.

Lemma 2.2[3]Strongly regular rings are duo rings.

Theorem 2.3R is a strongly regular ring if and only if R is a nonsingular ring,every principal left ideal of R is closed and a W-ideal.

ProofThe necessity follows at once from Lemma 2.1,the def i nition of regular rings and Lemma 2.2.

Now suppose that R is a nonsingular ring,every principal left ideal of R is closed and a W-ideal.

We prove that R is reduced f i rst.Let b∈R,b2=0,then there exists a closed left ideal K such that Rb⊕K≤eR.If K/=0,then for any k(/=0)∈K,by hypothesis Rk is a W-ideal which implies that there exists n＞0 such that kn/=0 and knR≤Rk.Let L be the left ideal generated by the set{kn|0/=k∈K,kn/=0,knR≤Rk}.Clearly,L≤K.Now for any k(/=0)∈K there exists kn(/=0)∈L such that kn∈Rk∩L,it follows that L≤eK. Thus,Rb⊕L≤eR.Now let cb∈Rb,kn∈L.Since kncb∈knR≤Rk,it follows that LRb?K∩Rb=0.Therefore,L?l(b).On the other hand,since b2=0,Rb?l(b).This implies that L⊕Rb≤l(b).Whence l(b)≤eR.Since R is nonsingular,b=0.If K=0,then Rb≤eR.It follows that Rb?l(b)≤eR,implying b=0.Hence R is reduced.

Now we prove that R is strongly regular.Let 0/=a∈R,then Ra≥Ra2.For ra∈Ra with ra/=0,where r∈R,it is clrar that Rra≤Ra.By hypothesis Rra is a W-ideal,hence there exists an integer n＞0 such that(ra)n/=0 and(ra)nR≤Rra,hence(ra)na∈Rra.If (ra)na=0,then(ra)n?1ra2=0,which implies(ra)n?1r∈l(a2).Since R is reduced,l(a2)=l(a). Thus(ra)n?1ra=0,hence(ra)n=0,a contradiction.Whence Rra∩Ra2?(ra)na/=0,which implies that Ra2is essential in Ra.And since Ra2is closed,it follows that Ra=Ra2,implying that there exits b∈R,such that a=ba2.Therefore R is strongly regular.This completes the proof.

Note that Theorem 2.3 is a generalization of[4,Theorem 1].The corollary below follows from Lemma 2.1 and Theorem 2.3.

Corollary 2.4R is a strongly regular ring if and only if R is a regular ring,every principal left ideal of R is a W-ideal.

The following lemma is due to Roger Yue Chi Ming[5].

Lemma 2.5The following are equivalent

(a)R is regular;

(b)Every R-module is p-injective.

Theorem 2.6R is a strongly regular ring if and only if R is left p-injective,every principal left ideal of R is a W-ideal,every simple left R-module is p-injective.

ProofThe necessity follows by Lemma 2.5 and Lemma 2.2.

Suppose the converse.Let 0/=a∈R with a2=0.Suppose that Ra is not simple,then Ra has maximal submodules.Let W be a maximal submodule of Ra,then Ra/W is simple.Let π:Ra→Ra/W be the canonical homomorphism.Since Ra/W is p-injective,π extends to a homomorphism f from R to Ra/W.Thus a+W=π(a)=f(a)=af(1)=aba+W,b∈R. Hence a?aba∈W.Since Ra is a W-ideal,there exists a integer n＞0 such that an/=0 and anb∈anR?Ra.Since a2=0,it follows that n=1,implying ab=ra∈Ra,r∈R. Thus(ab)a=(ra)a=0.Whence a∈W,which yields Ra=W,a contradiction.Hence Ra is simple.It follows that l(a)is a maximal left ideal of R.Def i ne g:Ra→R/l(a),by g(ra)=r+l(a),r∈R.It is easy to check that g is a homomorphism.Since R/l(a)is simple,R/l(a)is p-injective by hypothesis.Whence g extends to a homomorphism h from R to R/l(a).Thus,1+l(a)=g(a)=h(a)=ah(1)=ac+l(a),c∈R.Hence 1?ac∈l(a). Since Ra is a weak ideal,there exists n＞0 such that an/=0 and anR∈Ra.Hence anc∈Ra. From a2=0,it follows that n=1,implying ac∈Ra.Therefore,ac=ra,r∈R.Thus 0=(1?ac)a=(1?ra)a=a?ra2=a.Whence R is reduced.By[6,Theorem 6],it follows that R is a strongly regular ring.This completes the proof.

Theorem 2.7R is a strongly regular ring if and only if R is right p-injective,every principal left ideal of R is a W-ideal,every simple left R-module is p-injective.

ProofThe necessity is clear.

Suppose the converse.By the argument of Theorem 2.6,R is reduced.Because R is right p-injective,it follows that Ra=lr(a)by[7,Lemma 1.1].Since R is reduced,r(a)=r(a2).It follows that Ra=lr(a)=lr(a2)=Ra2.Hence a=ba2,b∈R.Thus R is strongly regular. This completes the proof.

According to Roger Yue Chi Ming[8],a left R-module M is called Tp-injective(two-sided ideal p-injective)if for any ideal I of R,r∈R,any left R-homomorphiam g:Ir→M,there exists y∈M such that g(tr)=try for all t∈I.

Obviously,Tp-injectivity implies p-injectivity.

In[8],the author proved that R a left and right self-injective strongly regular ring if and only if R is a left non-singular left Tp-injective ring such that every complement left ideal is an ideal if and only if R is a reduced left Tp-injective ring.We give the following theorem.

Theorem 2.8R is a nonsingular p-injective ring which every closed left ideal of R is aW-ideal if and only if R is a strongly regular ring.

ProofThe sufficiency is clear.

The necessity.Let a2=0,a∈R,then there exists a closed left ideal K of R,such that (Ra+l(a))⊕K≤eR.If K/=0,let 0/=k∈K,then by hypothesis K is a W-ideal which implies that there exists n＞0 such that kn/=0 and knR≤K.Let L be the left ideal generated by the set{kn|0/=k∈K,kn/=0,knR≤K}.By the same argument in Theorem 2.3,L≤eK and(Ra+l(a))⊕L≤eR.From the def i nition of L,it is evident that LRa?K,Ll(a)?K.Thus L(Ra+l(a))?K∩(Ra+l(a))=0.Since Ra2=0,Ra?l(a),implying Ll(a)=L(Ra+l(a))=0.Hence LRa=0 which implies La=0.Thus,L?l(a).Therefore (Ra+l(a))⊕L=l(a)≤eR.Since R is nonsingular,a=0.If K=0,then(Ra+l(a))≤eR. In this case,it is easy to see that a=0.Thus R is reduced.Hence R is strongly regular by[6, Theorem 6].This completes the proof of the theorem.

By Theorem 2.8 and[9,Theorem 2]we get

Corollary 2.9R is a semiprimitive p-injective ring which every closed left ideal of R is a W-ideal if and only if R is a strongly regular ring.

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tion:16D25,16D40

1002–0462(2014)04–0505–04

date:2012-12-21

Supported by the National Natural Science Foundation of China(11161006,11171142); Supported by the Natural Science Foundation of Guangxi Province(2011GXNSFA018144,018139,2010GXNSFB 013048,0991102);Supported by the Guangxi New Century 1000 Talents Project;Supported by the Guangxi Graduate Student Education Innovation Project(2011106030701M06);Supported by the SRF of Guangxi Education Committee

Biography:BAN Xiu-he(1962-),male,native of Pingguo,Guangxi,a lecturer of Guangxi Teacher’s Education University,M.S.D.,engages in theory of rings.

CLC number:O153.3Document code:A