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)={k∈Ml(R)∣RRkisprojective}.

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 [1]. In [2–6], the authors discussed the properties of left MC2 rings. In [1], a ring R is called left mininjective if rl(k)=kR for every k∈Ml(R), and R is said to be left minsymmetric if k∈Ml(R) always implies k∈Mr(R). According to [1], left mininjective ⇒left minsymmetric ⇒left MC2, and no reversal holds.

A ring R is called left universally mininjective [1] if Rk is an idempotent left ideal of R for every k∈Ml(R). The work in [2] 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 [3] if for each e∈MEl(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 e∈MEl(R) and g∈E(R).

Let F be a field and R={(ab0a)∣a,b∈F}. Then E(R)={(0000),(1001)} and MEl(R) is empty, so R is left WMC2. Now let S=(FF0F). Then MEl(S)={(1u00)∣u∈F} and E(S)={(0000),(1001),(1u00),(0u01)∣u∈F}. 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 [8] if aRe=0 implies eRa=0 for all a∈R and e∈E(R). Clearly, R is left idempotent reflexive if and only if for any a∈N(R) and e∈E(R), aRe=0 implies eRa=0 if and only if for any a∈J(R) and e∈E(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 [9], Osofsky proves that if R is a semiprime ring containing an injective maximal left ideal, then R is left self-injective. In [8], Kim and Baik prove that if R is left idempotent reflexive containing an injective maximal left ideal, then R is left self-injective. In [10], 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.

We start with the following theorem.

Theorem 1.

The following conditions are equivalent for a ring R:

R is left MC2;

for any a∈R and e∈MEl(R), eaRe=0 implies ea=0;

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

for any k,l∈Ml(R), kRl=0 implies lRk=0.

Proof.

(1) ⇒(2) Assume that a∈R and e∈MEl(R) with eaRe=0. If ea≠0, then Rea≅Re. By (1), Rea=Rg for some g∈MEl(R). Hence Rg=RgRg=ReaRea=R(eaRe)a=0, which is a contradiction. Hence ea=0.

(2) ⇒(3) Let e,g∈MEl(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,l∈Ml(R) with kRl=0. If lRk≠0, then RlRk=Rk. Hence Rk=RlRk=(Rl)2Rk, which implies Rl=Re for some e∈MEl(R). Since ReRk=RlRk=Rk≠0, there exists b∈R such that ebk≠0. Let g=e+ebk. Then g2=e+ebk+ebke+ebkebk=e+ebk=g∈MEl(R) because ebke∈RkRe=RkRl=0 and g≠0. 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 a∈Ml(R) and e∈MEl(R) with Ra≅Re. Then there exists g∈MEl(R) such that a=ga and l(a)=l(g). If (Ra)2=0, then RaR⊆l(a)=l(g), so aRg=0, by (4), gRa=0, which implies a=ga=0. It is a contradiction. Hence (Ra)2≠0, so Ra=Rh for some h∈MEl(R), which implies R is a left MC2 ring.

Corollary 2.

Left MC2 rings are left WMC2.

Proof.

Let e∈MEl(R) and g∈E(R) with gRe=0. If eRg≠0, then ebg≠0 for some b∈R. Clearly, ebg∈Ml(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 e∈MEl(R) and g∈E(R), eg≠0 implies gRe≠0;

for any e∈MEl(R), l(Re)∩E(R)⊆r(eR);

for any k∈MPl(R) and g∈E(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 g∈l(Re)∩E(R). Then gRe=0. We claim that eRg=0. Otherwise, there exists b∈R such that ebg≠0. Clearly, h=ebg+g-eg∈E(R) and eh=ebg≠0. By (2), we have hRe≠0. But hRe=0 because gRe=0. This is a contradiction. Hence eRg=0 and so g∈r(eR). Therefore l(Re)∩E(R)⊆r(eR).

(3) ⇒(4) Since k∈MPl(R), RRk is projective. It is easy to show that k=ek and l(k)=l(e) for some e∈MEl(R). Since gRk=0, gR⊆l(k). Therefore gRe=0, which implies g∈l(Re)∩E(R). By (3), eRg=0. Hence kRg=ekRg⊆eRg=0.

By Theorem 3, we have the following corollary.

Corollary 4.

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

(2) If R is a direct product of a family rings {Ri:i∈I}, 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:i∈I}, 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 ⋂i∈IAi=0. Let e∈MEl(R) and g∈E(R) satisfying gRe=0. For any i∈I, if e∈Ai, then eRg∈Ai; if e∉Ai, then we can easily show that ei=e+Ai∈MEl(Ri). Since Ri is a left WMC2 ring and giRiei=0, where gi=g+Ai, eiRigi=0. Hence eRg⊆Ai. In any case, we have eRg⊆Ai for all i∈I. Therefore eRg⊆⋂i∈IAi=0 and so eRg=0. This shows that R is a left WMC2 ring.

(2) Let e∈MEl(R) and g∈E(R) satisfying eg≠0. Clearly, in R̅=R/Zl(R), e̅=e+Zl(R)∈MEl(R̅), g̅=g+Zl(R)∈E(R̅). Since RReg≅RRe, eg∉Zl(R). Since R̅ is a left WMC2 ring, by Theorem 3, g̅R̅e̅≠0, which implies gRe≠0. 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 [3] 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 e∈MEl(R) and a∈R satisfying eaRe=0. Set g=1-e+ea. Then, clearly, g∈E(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 [8] if aRe=0 implies eRa=0 for all a∈R and e∈E(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 [8] 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 [10] 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=M⊕N for some minimal left ideal N of R. Hence we have M=Re and N=R(1-e) for some e2=e∈R. If MN=0, then eR(1-e)=0. Since R is left WMC2 and 1-e∈MEl(R), (1-e)Re=0. So e is central. Now let L be any proper essential left ideal of R and f:L→N any nonzero left R-homomorphism. Then L/U≅N, where U=kerf is a maximal submodule of L. Now L=U⊕V, where V≅N=R(1-e) is a minimal left ideal of R. Since e is central, V=R(1-e). For any z∈L, let z=x+y, where x∈U, y∈V. 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)x∈V∩U=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 MN≠0, by the proof of [11, Proposition 5], we have NR is injective. Hence R=M⊕N is left self-injective.

A ring R is called strongly left DS [3] if k2≠0 for all k∈Ml(R). Since strongly left DS⇒left DS⇒left mininjective ⇒left minsymmetric ⇒left MC2⇒left 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 [5] if for any a∈N(R), rl(a)=aR, and R is said to be left NC2 [5] if for any a∈N(R), RaR is projective implies that Ra=Re for some e∈E(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 [12] 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 [13] 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 k∈Ml(R), rl(k)=kR⊕Xk, 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 e∈MEl(R) and g∈E(R) satisfying eg≠0. Since R is a left min-AP-injective ring and l(e)=l(eg), eR=rl(e)=rl(eg)=egR⊕Xeg, where Xeg is a right ideal R. Set e=egb+x, b∈R and x∈Xeg. Then eg=e(eg)=egbeg+xeg, so xeg=eg-egbeg∈egR∩Xeg, which implies xeg=0, so eg=egbeg. Let h=egb. Then h∈MEl(R) and egR=hR. Therefore hR=hRhR=egRegR which implies gRe≠0. By Theorem 3, R is a left WMC2 ring.

(2) Assume that e∈MEl(R) and a∈R satisfying eaRe=0. If ea≠0, then ea∈Soc(RR)⊆Soc(RR). Thus there exists a minimal right ideal kR of R such that kR⊆eaR. Clearly, l(k)=l(ea)=l(e) and kRkR⊆eaReaR=0. Hence RkR⊆l(k). Let I be a complement right ideal of RkR in R. Then I⊆l(k) and e∈Soc(RR)⊆Soc(RR)⊆RkR⊕I⊆l(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,c∈R;

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

symmetric if abc=0 implies acb=0 for all a,b,c∈R;

ZC [16] if ab=0 implies ba=0 for all a,b∈R;

ZI [16] if ab=0 implies aRb=0 for all a,b∈R.

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.

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 [17], a ring R is said to be left HI if R is left hereditary containing an injective maximal left ideal. Osofsky [9] 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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