IJMMSInternational Journal of Mathematics and Mathematical Sciences1687-04250161-1712Hindawi Publishing Corporation29430110.1155/2011/294301294301Research ArticleLeft WMC2 RingsWeiJunchao1WernerFrank1School of MathematicsYangzhou UniversityYangzhou 225002Chinayzu.edu.cn2011306201120111201201103052011090520112011Copyright © 2011 Junchao Wei.This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We introduce in this paper the concept of left WMC2 rings and concern ourselves with rings containing an injective maximal left ideal. Some known results for left idempotent reflexive rings and left HI rings can be extended to left WMC2 rings. As applications, we are able to give some new characterizations of regular left self-injective rings with nonzero socle and extend some known results on strongly regular rings.

Throughout this paper, R denotes an associative ring with identity, and all modules are unitary. For any nonempty subset X of a ring R, r(X)=rR(X) and l(X)=lR(X) denote the set of right annihilators of X and the set of left annihilators of X, respectively. We use J(R), N*(R), N(R), Zl(R), E(R), Soc(RR), and Soc(RR) for the Jacobson radical, the prime radical, the set of all nilpotent elements, the left singular ideal, the set of all idempotent elements, the left socle, and the right socle of R, respectively.

An element k of R is called left minimal if Rk is a minimal left ideal. An element e of R is called left minimal idempotent if e2=e is left minimal. We use Ml(R) and MEl(R) for the set of all left minimal elements and the set of all left minimal idempotent elements of R, respectively. Moreover, let MPl(R)={kMl(R)RRk  is  projective}.

A ring R is called left MC2 if every minimal left ideal which is isomorphic to a summand of RR is a summand. Left MC2 rings were initiated by Nicholson and Yousif in . In , the authors discussed the properties of left MC2 rings. In , a ring R is called left mininjective if rl(k)=kR for every kMl(R), and R is said to be left minsymmetric if kMl(R) always implies kMr(R). According to , left mininjective left minsymmetric left MC2, and no reversal holds.

A ring R is called left universally mininjective  if Rk is an idempotent left ideal of R for every kMl(R). The work in  uses the term left DS for the left universally mininjective. According to [1, Lemma  5.1], left DS rings are left mininjective.

A ring R is called left min-abel  if for each eMEl(R), e is left semicentral in R, and R is said to be strongly left min-abel [3, 7] if every element of MEl(R) is central in R.

A ring R is called left WMC2 if gRe=0 implies eRg=0 for eMEl(R) and gE(R).

Let F be a field and R={(ab0a)a,bF}. Then E(R)={(0000),(1001)} and MEl(R) is empty, so R is left WMC2. Now let S=(FF0F). Then MEl(S)={(1u00)uF  } and E(S)={(0000),(1001),(1u00),(0u01)uF}. Since (0u01)S(1000)=0 and (1000)S(0u01)0, S is not left WMC2.

Let R be any ring and S1=R[x] and S2=R[[x]]. Then MEl(S1) and MEl(S2) are all empties, so S1 and S2 are all left WMC2.

A ring R is called left idempotent reflexive  if aRe=0 implies eRa=0 for all aR and eE(R). Clearly, R is left idempotent reflexive if and only if for any aN(R) and eE(R), aRe=0 implies eRa=0 if and only if for any aJ(R) and eE(R), aRe=0 implies eRa=0. Therefore, left idempotent reflexive rings are left WMC2.

In general, the existence of an injective maximal left ideal in a ring R can not guarantee the left self-injectivity of R. In , Osofsky proves that if R is a semiprime ring containing an injective maximal left ideal, then R is left self-injective. In , Kim and Baik prove that if R is left idempotent reflexive containing an injective maximal left ideal, then R is left self-injective. In , Wei and Li prove that if R is left MC2 containing an injective maximal left ideal, then R is left self-injective. Motivated by these results, in this paper, we show that if R is a left WMC2 ring containing an injective maximal left ideal, then R is left self-injective. As an application of this result, we show that a ring R is a semisimple Artinian ring if and only if R is a left WMC ring and left HI ring.

Theorem 1.

The following conditions are equivalent for a ring R:

R is left MC2;

for any aR and eMEl(R), eaRe=0 implies ea=0;

for any e,gMEl(R), (g-e)Re=0 implies that e=eg;

for any k,lMl(R), kRl=0 implies lRk=0.

Proof.

(1) (2) Assume that aR and eMEl(R) with eaRe=0. If ea0, then ReaRe. By (1), Rea=Rg for some gMEl(R). Hence Rg=RgRg=ReaRea=R(eaRe)a=0, which is a contradiction. Hence ea=0.

(2) (3) Let e,gMEl(R) such that (g-e)Re=0. Then e(g-e)Re=0. By (2), e(g-e)=0. Hence e=eg.

(3) (4) Assume that k,lMl(R) with kRl=0. If lRk0, then RlRk=Rk. Hence Rk=RlRk=(Rl)2Rk, which implies Rl=Re for some eMEl(R). Since ReRk=RlRk=Rk0, there exists bR such that ebk0. Let g=e+ebk. Then g2=e+ebk+ebke+ebkebk=e+ebk=gMEl(R) because ebkeRkRe=RkRl=0 and g0. Since (g-e)Re=ebkRe=ebkRl=0, by (3), e=eg. Hence g=eg=e, which implies ebk=0. It is a contradiction. Therefore lRk=0.

(4) (1) Let aMl(R) and eMEl(R) with RaRe. Then there exists gMEl(R) such that a=ga and l(a)=l(g). If (Ra)2=0, then RaRl(a)=l(g), so aRg=0, by (4), gRa=0, which implies a=ga=0. It is a contradiction. Hence (Ra)20, so Ra=Rh for some hMEl(R), which implies R is a left MC2 ring.

Corollary 2.

Left MC2 rings are left WMC2.

Proof.

Let eMEl(R) and gE(R) with gRe=0. If eRg0, then ebg0 for some bR. Clearly, ebgMl(R) and (ebg)Re=0. Since R is a left MC2 ring, by Theorem 1, eR(ebg)=0, which implies ebg=0, and this is a contradiction. Hence eRg=0 and so R is a left WMC2 ring.

We do not know whether the converse of Corollary 2 holds. However, we have the following characterization of left WMC2 rings.

Theorem 3.

Let R be a ring. Then the following conditions are equivalent:

R is a left WMC2 ring;

for any eMEl(R) and gE(R), eg0 implies gRe0;

for any eMEl(R), l(Re)E(R)r(eR);

for any kMPl(R) and gE(R), gRk=0 implies kRg=0.

Proof.

(4) (1) (2) It is easy to show by the definition of left WMC2 ring.

(2) (3) Let gl(Re)E(R). Then gRe=0. We claim that eRg=0. Otherwise, there exists bR such that ebg0. Clearly, h=ebg+g-egE(R) and eh=ebg0. By (2), we have hRe0. But hRe=0 because gRe=0. This is a contradiction. Hence eRg=0 and so gr(eR). Therefore l(Re)E(R)r(eR).

(3) (4) Since kMPl(R), RRk is projective. It is easy to show that k=ek and l(k)=l(e) for some eMEl(R). Since gRk=0, gRl(k). Therefore gRe=0, which implies gl(Re)E(R). By (3), eRg=0. Hence kRg=ekRgeRg=0.

By Theorem 3, we have the following corollary.

Corollary 4.

(1) Let R be a left WMC2 ring. If eE(R) satisfying ReR=R, then eRe is left WMC2.

(2) If R is a direct product of a family rings {Ri:iI}, then R is a left WMC2 ring if and only if every Ri is left WMC2.

Theorem 5.

(1) If R is a subdirect product of a family left WMC2 rings {Ri:iI}, then R is a left WMC2 ring.

(2) If R/Zl(R) is a left WMC2 ring, so is R.

Proof.

(1) Let Ri=R/Ai, where Ai are ideals of R with iIAi=0. Let eMEl(R) and gE(R) satisfying gRe=0. For any iI, if eAi, then eRgAi; if eAi, then we can easily show that ei=e+AiMEl(Ri). Since Ri is a left WMC2 ring and giRiei=0, where gi=g+Ai, eiRigi=0. Hence eRgAi. In any case, we have eRgAi for all iI. Therefore eRgiIAi=0 and so eRg=0. This shows that R is a left WMC2 ring.

(2) Let eMEl(R) and gE(R) satisfying eg0. Clearly, in R̅=R/Zl(R), e̅=e+Zl(R)MEl(R̅), g̅=g+Zl(R)E(R̅). Since RRegRRe, egZl(R). Since R̅ is a left WMC2 ring, by Theorem 3, g̅R̅e̅0, which implies gRe0. Thus R is a left WMC2 ring by Theorem 3.

Theorem 6.

(1)  R is a strongly left min-abel ring if and only if R is a left min-abel left WMC2 ring.

(2) If R/Zl(R) is a strongly left min-abel ring, then so is R.

Proof.

(1) Theorem  1.8 in  shows that R is a strongly left min-abel ring if and only if R is a left min-abel left MC2 ring, so by Corollary 2, we obtain that strongly left min-abel ring is left min-abel left WMC2.

Conversely, let R be a left min-abel left WMC2 ring. Let eMEl(R) and aR satisfying eaRe=0. Set g=1-e+ea. Then, clearly, gE(R) and eg=ea. Since R is a left min-abel ring, (1-e)Re=(1-e)eRe=0, so gRe=0. Since R is a left WMC2 ring, eRg=0, which implies ea=eg=0, by Theorem 1, R is a left MC2 ring. Hence R is a strongly left min-abel ring.

(2) It is an immediate corollary of (1), [3, Corollary  1.5(2)] and Theorem 5(2).

A ring R is called left idempotent reflexive  if aRe=0 implies eRa=0 for all aR and eE(R). Clearly, left idempotent reflexive rings are left WMC2.

In general, the existence of an injective maximal left ideal in a ring R cannot guarantee the left self-injectivity of R. Proposition  5 in  proves that if R is a left idempotent reflexive ring containing an injective maximal left ideal, then R is a left self-injective ring. Theorem  4.1 in  proves that if R is a left MC2 ring containing an injective maximal left ideal, then R is a left self-injective ring. We can generalize the results as follows.

Theorem 7.

Let R be a left WMC2 ring. If R contains an injective maximal left ideal, then R is a left self-injective ring.

Proof.

Let M be an injective maximal left ideal of R. Then R=MN for some minimal left ideal N of R. Hence we have M=Re and N=R(1-e) for some e2=eR. If MN=0, then eR(1-e)=0. Since R is left WMC2 and 1-eMEl(R), (1-e)Re=0. So e is central. Now let L be any proper essential left ideal of R and f:LN any nonzero left R-homomorphism. Then L/UN, where U=kerf is a maximal submodule of L. Now L=UV, where VN=R(1-e) is a minimal left ideal of R. Since e is central, V=R(1-e). For any zL, let z=x+y, where xU, yV. Then f(z)=f(x)+f(y)=f(y). Since y=y(1-e)=(1-e)y, f(z)=f(y)=f(y(1-e))=yf(1-e). Since x(1-e)=(1-e)xVU=0, xf(1-e)=fx(1-e)=f(0)=0. Thus f(z)=yf(1-e)=yf(1-e)+xf(1-e)=(y+x)f(1-e)=zf(1-e). Hence RN is injective. If MN0, by the proof of [11, Proposition  5], we have NR is injective. Hence R=MN is left self-injective.

A ring R is called strongly left DS  if k20 for all kMl(R). Since strongly left DSleft DSleft mininjective left minsymmetric left MC2left WMC2 and strongly left min-abel left WMC2, we have the following corollary.

Corollary 8.

Let R contain an injective maximal left ideal. If R satisfies one of the following conditions, then R is a left self-injective ring.

R is a strongly left DS ring.

R is a left DS ring.

R is a left mininjective ring.

R is a left minsymmetric ring.

R is a strongly left min-abel ring.

R is a left MC2 ring.

It is well known that if R is a left self-injective ring, then J(R)=Zl(R). Therefore by [2, Theorem  5.1] and Corollary 8, we have the following corollary.

Corollary 9.

Let R contain an injective maximal left ideal. Then R is left self-injective if and only if J(R)=Zl(R).

A ring R is called left nil-injective  if for any aN(R), rl(a)=aR, and R is said to be left NC2  if for any aN(R), RaR is projective implies that Ra=Re for some eE(R). By [5, Theorem  2.22], left nil-injective rings are left NC2 and left NC2 rings are left MC2. A ring R is right Kasch if every simple right R-module can be embedded in RR, and R is said to be left C2  if every left ideal that is isomorphic to a direct summand of RR is itself a direct summand. Clearly, left self-injective rings are left C2  and left C2 rings are left NC2 and by [14, Lemma  1.15], right Kasch rings are left C2. Hence, we have the following corollary.

Corollary 10.

(1) Let R contain an injective maximal left ideal. Then the following conditions are equivalent:

R is a left self-injective ring;

R is a left nil-injective ring;

R is a left C2 ring;

R is a left NC2 ring.

(2) If R is a right Kasch ring containing an injective maximal left ideal, then R is a left self-injective ring.

A ring R is called left min-AP-injective if for any kMl(R), rl(k)=kRXk, where Xk is a right ideal of R. Clearly, left mininjective rings are left min AP-injective.

Lemma 11.

(1) If R is a left min -AP-injective ring, then R is left WMC2.

(2) If Soc(RR)Soc(RR), then R is left WMC2.

Proof.

(1) Let eMEl(R) and gE(R) satisfying eg0. Since R is a left min-AP-injective ring and l(e)=l(eg), eR=rl(e)=rl(eg)=egRXeg, where Xeg is a right ideal R. Set e=egb+x, bR and xXeg. Then eg=e(eg)=egbeg+xeg, so xeg=eg-egbegegRXeg, which implies xeg=0, so eg=egbeg. Let h=egb. Then hMEl(R) and egR=hR. Therefore hR=hRhR=egRegR which implies gRe0. By Theorem 3, R is a left WMC2 ring.

(2) Assume that eMEl(R) and aR satisfying eaRe=0. If ea0, then eaSoc(RR)Soc(RR). Thus there exists a minimal right ideal kR of R such that kReaR. Clearly, l(k)=l(ea)=l(e) and kRkReaReaR=0. Hence RkRl(k). Let I be a complement right ideal of RkR in R. Then Il(k) and eSoc(RR)Soc(RR)RkRIl(k)=l(e), which is a contradiction. Hence ea=0. By Theorem 1, R is a left MC2 ring, so R is left WMC2 by Corollary 2.

Since left mininjective rings are left min-AP-injective and Soc(RR)Soc(RR). Hence by Theorem 7, Corollary 8 and Lemma 11, we have the following theorem.

Theorem 12.

Let R contain an injective maximal left ideal. Then the following conditions are equivalent:

R is left self-injective;

R is left min-AP-injective;

Soc(RR)Soc(RR).

A ring R is called

strongly reflexive if aRbRc=0 implies aRcRb=0 for all a,b,cR;

reflexive [8, 15] if aRb=0 implies bRa=0 for all a,bR;

symmetric if abc=0 implies acb=0 for all a,b,cR;

ZC  if ab=0 implies ba=0 for all a,bR;

ZI  if ab=0 implies aRb=0 for all a,bR.

Evidently, we have the following proposition.

Proposition 13.

(1) The following conditions are equivalent for a ring R:

R is semiprime;

R is strongly reflexive and every proper essential right ideal of R contains no nonzero nilpotent ideal;

R is reflexive and every proper essential right ideal of R contains no nonzero nilpotent ideal;

R is strongly reflexive and N*(R)Zl(R)=0;

R is reflexive and N*(R)Zl(R)=0.

(2)R is symmetric if and only if R is ZI and strongly reflexive.

(3)R is reversible if and only if R is ZI and reflexive.

(4) Strongly reflexive reflexive left idempotent reflexive.

It is well known that if R is a left self-injective ring, then Zl(R)=J(R), so R/Zl(R) is semiprimitive. Hence R/Zl(R) is left WMC2 by Proposition 13. Thus, by Theorems 5 and 7, we have the following theorem.

Theorem 14.

Let R contain an injective maximal left ideal. Then

R is a left self-injective ring if and only if R/Zl(R) is a left WMC2 ring.

If R satisfies one of the following conditions, then R is a left self-injective:

R is a semiprime ring;

R is a strongly reflexive;

R is reflexive;

R is a left idempotent reflexive.

Recall that a ring R is left pp if every principal left ideal of R is projective as left R-module. As an application of Theorem 7, we have the following result.

Theorem 15.

The following conditions are equivalent for a ring R:

R is a von Neumann regular left self-injective ring with Soc(RR)0;

R is a left WMC2 left pp ring containing an injective maximal left ideal;

R is a left pp ring containing an injective maximal left ideal and R/Zl(R) is a left WMC2 ring.

Proof.

(1) (3) is trivial.

(3) (2) is a direct result of Theorem 5(2).

(2) (1) By Theorem 7, R is left self-injective. Hence, by [13, Theorem  1.2], R is left C2, so R is von Neumann regular because R is left pp. In addition, we have Soc(RR)0 since there is an injective maximal left ideal.

By , a ring R is said to be left HI if R is left hereditary containing an injective maximal left ideal. Osofsky  proves that a left self-injective left hereditary ring is semisimple Artinian. We can generalize the result as follows.

Corollary 16.

The following conditions are equivalent for a ring R:

R is semisimple Artinian;

R is left WMC2 left HI;

R is left NC2 left HI;

R is left min-AP-injective left HI;

R is left idempotent reflexive left HI.

Acknowlegment

Project supported by the Foundation of Natural Science of China (10771182, 10771183).

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