Left

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Volume 2011, Article ID 294301,8pages doi:10.1155/2011/294301

Research Article

Left

WMC

2 Rings

Junchao Wei

School of Mathematics, Yangzhou University, Yangzhou 225002, China

Correspondence should be addressed to Junchao Wei,[email protected]

Received 12 January 2011; Revised 3 May 2011; Accepted 9 May 2011

Academic Editor: Frank Werner

Copyrightq2011 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 leftWMC2 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 subsetXof a ringR,rX rRXandlX lRXdenote the set of right annihilators ofX and the set of left annihilators ofX, respectively. We useJR,

N_∗R,NR,ZlR,ER,Soc_RR, andSocRRfor 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 ofR, respectively.

An elementkofRis called left minimal ifRkis a minimal left ideal. An elementeof

Ris called left minimal idempotent ife2 _e_{is left minimal. We use}_M_l_R_and_ME_l_R_for

the set of all left minimal elements and the set of all left minimal idempotent elements ofR, respectively. Moreover, letMPlR {k∈MlR| RR k 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. LeftMC2 rings were initiated by Nicholson and Yousif in 1. In2–6, the authors discussed the properties of leftMC2 rings. In1, a ringRis called left mininjective ifrlk kRfor everyk ∈MlR, andRis said to be left minsymmetric if

k∈MlRalways impliesk∈MrR. According to1, left mininjective⇒left minsymmetric ⇒leftMC2, and no reversal holds.

A ring R is called left universally mininjective 1 ifRk is an idempotent left ideal of R for every k ∈ MlR. The work in 2 uses the term left DS for the left universally mininjective. According to1, Lemma 5.1, leftDSrings are left mininjective.

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A ringRis called leftWMC2 ifgRe0 implieseRg0 fore∈MElRandg ∈ER. LetFbe a field andR {a b₀_a | a, b ∈F}. ThenER {0 0

0 0

,1 0 0 1

}andMElR

is empty, soR is leftWMC2. Now letS F F₀ _F. Then MElS {1u 0 0

| u ∈ F }and

ES {0 0 0 0

,1 0 0 1

,1u 0 0

,0u 0 1

|u∈F}. Since0u 0 1

S1 0 0 0

0 and1 0 0 0

S0u 0 1

/

0,Sis not leftWMC2.

LetRbe any ring andS1 RxandS2 Rx. ThenMElS1andMElS2are all

empties, soS1andS2are all leftWMC2.

A ringRis called left idempotent reflexive8ifaRe0 implieseRa0 for alla∈R and e ∈ ER. Clearly,R is left idempotent reflexive if and only if for anya ∈ NRand

e ∈ ER,aRe 0 implieseRa 0 if and only if for any a∈ JRand e ∈ ER,aRe 0 implieseRa0. Therefore, left idempotent reflexive rings are leftWMC2.

In general, the existence of an injective maximal left ideal in a ringRcan not guarantee the left self-injectivity ofR. In9, Osofsky proves that ifR is a semiprime ring containing an injective maximal left ideal, thenR is left self-injective. In8, Kim and Baik prove that ifRis left idempotent reflexive containing an injective maximal left ideal, thenRis left self-injective. In10, Wei and Li prove that ifRis leftMC2 containing an injective maximal left ideal, thenRis left self-injective. Motivated by these results, in this paper, we show that ifR is a leftWMC2 ring containing an injective maximal left ideal, thenRis left self-injective. As an application of this result, we show that a ringRis a semisimple Artinian ring if and only ifRis a leftWMCring and leftHIring.

We start with the following theorem.

Theorem 1. The following conditions are equivalent for a ringR:

1Ris leftMC2;

2for anya∈Rande∈MElR,eaRe0 impliesea0;

3for anye, g∈MElR,g−eRe0 implies thateeg;

4for anyk, l∈MlR,kRl0 implieslRk0.

Proof. 1⇒2Assume thata∈Rande∈MElRwitheaRe0. Ifea /0, thenRea∼Re. By1,ReaRgfor someg ∈MElR. HenceRgRgRgReaReaReaRea0, which is a contradiction. Henceea0.

2⇒3Let e, g ∈ MElR such that g −eRe 0. Theneg −eRe 0. By 2,

eg−e 0. Henceeeg.

3⇒4Assume thatk, l∈MlRwithkRl0. IflRk /0, thenRlRkRk. HenceRk

RlRk Rl2_Rk_{, which implies}_Rl_Re_{for some}_e_∈_ME_l_R_{. Since}_ReRk_RlRk_{Rk /}_0,

there existsb∈Rsuch thatebk /0. Letgeebk. Theng2_e_ebk_ebke_ebkebk_e_ebk

g∈MElRbecauseebke∈RkReRkRl0 andg /0. Sinceg−eReebkReebkRl0, by3,e eg. Hence g eg e, which impliesebk 0. It is a contradiction. Therefore

lRk0.

4⇒1Leta∈MlRande∈MElRwithRa∼Re. Then there existsg ∈MElR such thata gaandla lg. If Ra2 0, thenRaR⊆ la lg, soaRg 0, by 4,

gRa0, which impliesaga0. It is a contradiction. HenceRa2/0, soRaRhfor some

h∈MElR, which impliesRis a leftMC2 ring.

Corollary 2. LeftMC2 rings are leftWMC2.

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which impliesebg 0, and this is a contradiction. HenceeRg 0 and soRis a leftWMC2 ring.

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

Theorem 3. LetRbe a ring. Then the following conditions are equivalent:

1Ris a leftWMC2 ring;

2for anye∈MElRandg ∈ER,eg /0 impliesgRe /0;

3for anye∈MElR,lRe∩ER⊆reR;

4for anyk∈MPlRandg ∈ER,gRk0 implieskRg 0.

Proof. 4⇒1⇒2It is easy to show by the definition of leftWMC2 ring.

2⇒3Letg ∈lRe∩ER. ThengRe0. We claim thateRg 0. Otherwise, there existsb∈Rsuch thatebg /0. Clearly,h ebgg−eg ∈ERandehebg /0. By2, we havehRe /0. ButhRe 0 becausegRe 0. This is a contradiction. HenceeRg 0 and so

g∈reR. ThereforelRe∩ER⊆reR.

3⇒4 Sincek ∈ MPlR, RR k is projective. It is easy to show thatk ek and

lk lefor somee∈MElR. SincegRk0,gR⊆lk. ThereforegRe0, which implies

g∈lRe∩ER. By3,eRg 0. HencekRgekRg⊆eRg0.

ByTheorem 3, we have the following corollary.

Corollary 4. 1Let Rbe a leftWMC2 ring. Ife ∈ ERsatisfyingReR R, then eRe is left

WMC2.

2IfRis a direct product of a family rings{Ri :i∈I}, thenRis a leftWMC2 ring if and

only if everyRiis leftWMC2.

Theorem 5. 1IfRis a subdirect product of a family leftWMC2 rings{Ri:i∈I}, thenRis a left

WMC2 ring.

2IfR/ZlRis a leftWMC2 ring, so isR.

Proof. 1 LetRi R/Ai, where Ai are ideals ofR with _i∈IAi 0. Let e ∈ MElRand

g ∈ERsatisfyinggRe 0. For anyi∈I, ife ∈ Ai, theneRg ∈Ai; ife /∈ Ai, then we can

easily show thatei eAi ∈MElRi. SinceRiis a leftWMC2 ring andgiRiei 0, where

gigAi,eiRigi0. HenceeRg ⊆Ai. In any case, we haveeRg ⊆Aifor alli∈I. Therefore

eRg⊆_i∈IAi0 and soeRg 0. This shows thatRis a leftWMC2 ring.

2 Let e ∈ MElR and g ∈ ERsatisfying eg /0. Clearly, in R R/ZlR,e

eZlR ∈MElR,g gZlR∈ ER. Since RR eg∼RRe,eg /∈ ZlR. SinceRis a left

WMC2 ring, byTheorem 3,gRe /0, which impliesgRe /0. ThusRis a leftWMC2 ring by

Theorem 3.

Theorem 6. 1Ris a strongly left min-abel ring if and only ifRis a left min-abel leftWMC2 ring.

2IfR/ZlRis a strongly left min-abel ring, then so isR.

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Conversely, letR be a left min-abel left WMC2 ring. Lete ∈ MElR and a ∈ R satisfyingeaRe 0. Setg 1−eea. Then, clearly,g ∈ERandeg ea. SinceRis a left min-abel ring,1−eRe 1−eeRe0, sogRe0. SinceRis a leftWMC2 ring,eRg 0, which impliesea eg 0, byTheorem 1,R is a leftMC2 ring. HenceR is a strongly left min-abel ring.

2It is an immediate corollary of1,3, Corollary 1.52andTheorem 52.

A ringRis called left idempotent reflexive8ifaRe0 implieseRa0 for alla∈R ande∈ER. Clearly, left idempotent reflexive rings are leftWMC2.

In general, the existence of an injective maximal left ideal in a ringRcannot guarantee the left self-injectivity ofR. Proposition 5 in8proves that ifRis a left idempotent reflexive ring containing an injective maximal left ideal, thenRis a left self-injective ring. Theorem 4.1 in10proves that ifRis a left MC2 ring containing an injective maximal left ideal, thenRis a left self-injective ring. We can generalize the results as follows.

Theorem 7. LetRbe a leftWMC2 ring. IfRcontains an injective maximal left ideal, thenRis a left

self-injective ring.

Proof. LetMbe an injective maximal left ideal ofR. ThenRM⊕Nfor some minimal left idealNofR. Hence we haveMReandNR1−efor somee2_e_∈_R_{. If}_MN_{0, then}

eR1−e 0. SinceRis leftWMC2 and 1−e∈MElR,1−eRe0. Soeis central. Now let

Lbe any proper essential left ideal ofRandf :L → Nany nonzero leftR−homomorphism. ThenL/U ∼ N, whereU kerf is a maximal submodule ofL. NowL U⊕V, where

V ∼NR1−eis a minimal left ideal ofR. Sinceeis central,V R1−e. For anyz∈L, let

zxy, wherex∈U,y∈V. Thenfz fx fy fy. Sinceyy1−e 1−ey,

fz fy fy1−e yf1−e. Sincex1−e 1−ex∈ V ∩U 0,xf1−e

fx1−e f0 0. Thusfz yf1−e yf1−exf1−e yxf1−e zf1−e. HenceRNis injective. IfMN /0, by the proof of11, Proposition 5, we have RNis injective.

HenceRM⊕Nis left self-injective.

A ringRis called strongly leftDS3if k2_/_{0 for all}_k _∈ _M_l_R_{. Since strongly left}

DS ⇒ left DS ⇒ left mininjective⇒ left minsymmetric ⇒ left MC2 ⇒ left WMC2 and strongly left min-abel⇒leftWMC2, we have the following corollary.

Corollary 8. Let R contain an injective maximal left ideal. If R satisfies one of the following

conditions, thenRis a left self-injective ring.

1Ris a strongly leftDSring.

2Ris a leftDSring.

3Ris a left mininjective ring.

4Ris a left minsymmetric ring.

5Ris a strongly left min-abel ring.

6Ris a leftMC2 ring.

It is well known that ifRis a left self-injective ring, thenJR ZlR. Therefore by2, Theorem 5.1andCorollary 8, we have the following corollary.

Corollary 9. LetRcontain an injective maximal left ideal. ThenRis left self-injective if and only if

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A ringRis called leftnil-injective5if for anya∈NR,rla aR, andRis said to be leftNC25if for anya∈NR,RRais projective implies thatRaRefor somee∈ER.

By5, Theorem 2.22, leftnil-injective rings are leftNC2 and leftNC2 rings are leftMC2. A ringRis right Kasch if every simple rightR-module can be embedded inRR, andRis said to

be leftC212if every left ideal that is isomorphic to a direct summand ofRRis itself a direct

summand. Clearly, left self-injective rings are leftC213and leftC2 rings are leftNC2 and by14, Lemma 1.15, right Kasch rings are leftC2. Hence, we have the following corollary.

Corollary 10. 1LetRcontain an injective maximal left ideal. Then the following conditions are

equivalent:

aRis a left self-injective ring;

bRis a left nil-injective ring;

cRis a leftC2 ring;

dRis a leftNC2 ring.

2IfRis a right Kasch ring containing an injective maximal left ideal, thenRis a left self-injective ring.

A ringRis called left min-AP-injective if for anyk ∈MlR,rlk kR⊕Xk, where

Xkis a right ideal ofR. Clearly, left mininjective rings are left minAP-injective.

Lemma 11. 1IfRis a left min -AP-injective ring, thenRis leftWMC2.

2IfSoc_RR⊆SocRR, thenRis leftWMC2.

Proof. 1Lete ∈MElRandg ∈ERsatisfyingeg /0. SinceRis a left min-AP-injective ring andle leg,eR rle rleg egR⊕Xeg, whereXeg is a right idealR. Sete

egbx,b∈Randx∈Xeg. Thenegeeg egbegxeg, soxegeg−egbeg∈egR∩Xeg,

which implies xeg 0, so eg egbeg. Leth egb. Then h ∈ MElRand egR hR. ThereforehR hRhR egRegR which impliesgRe /0. ByTheorem 3,R is a leftWMC2 ring.

2 Assume that e ∈ MElRand a ∈ R satisfying eaRe 0. If ea /0, then ea ∈

SocRR ⊆ SocRR. Thus there exists a minimal right idealkRof R such thatkR ⊆ eaR.

Clearly,lk lea leandkRkR⊆eaReaR0. HenceRkR⊆lk. LetIbe a complement right ideal ofRkRinR. ThenI ⊆ lkande ∈Soc_RR ⊆SocRR ⊆RkR⊕I ⊆ lk le, which is a contradiction. Henceea0. ByTheorem 1,Ris a leftMC2 ring, soRis leftWMC2

byCorollary 2.

Since left mininjective rings are left min-AP-injective andSoc_RR⊆ SocRR. Hence

byTheorem 7,Corollary 8andLemma 11, we have the following theorem.

Theorem 12. Let R contain an injective maximal left ideal. Then the following conditions are

equivalent:

1Ris left self-injective;

2Ris left min-AP-injective;

3Soc_RR⊆SocRR.

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1strongly reflexive ifaRbRc0 impliesaRcRb0 for alla, b, c∈R;

2reflexive8,15ifaRb0 impliesbRa0 for alla, b∈R;

3symmetric ifabc0 impliesacb0 for alla, b, c∈R;

4ZC16ifab0 impliesba0 for alla, b∈R;

5ZI16ifab0 impliesaRb0 for alla, b∈R.

Evidently, we have the following proposition.

Proposition 13. 1The following conditions are equivalent for a ringR:

aRis semiprime;

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

cRis reflexive and every proper essential right ideal ofRcontains no nonzero nilpotent ideal;

dRis strongly reflexive andN∗R∩ZlR 0;

eRis reflexive andN∗R∩ZlR 0.

2Ris symmetric if and only ifRisZIand strongly reflexive.

3Ris reversible if and only ifRisZIand reflexive.

4Strongly reflexive⇒reflexive⇒left idempotent reflexive.

It is well known that ifRis a left self-injective ring, thenZlR JR, soR/ZlRis semiprimitive. HenceR/ZlRis leftWMC2 byProposition 13. Thus, by Theorems5and7, we have the following theorem.

Theorem 14. LetRcontain an injective maximal left ideal. Then

1Ris a left self-injective ring if and only ifR/ZlRis a leftWMC2 ring.

2IfRsatisfies one of the following conditions, thenRis a left self-injective:

aRis a semiprime ring;

bRis a strongly reflexive;

cRis reflexive;

dRis 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 ofTheorem 7, we have the following result.

Theorem 15. The following conditions are equivalent for a ringR:

1Ris a von Neumann regular left self-injective ring withSoc_RR/0;

2Ris a leftWMC2 leftppring containing an injective maximal left ideal;

3Ris a leftppring containing an injective maximal left ideal andR/ZlRis a leftWMC2 ring.

Proof. 1⇒3is trivial.

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2⇒1ByTheorem 7,Ris left self-injective. Hence, by13, Theorem 1.2,Ris leftC2, so R is von Neumann regular becauseR is left pp. In addition, we have Soc_RR/0 since there is an injective maximal left ideal.

By 17, a ringR is said to be leftHI 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 ringR:

1Ris semisimple Artinian;

2Ris leftWMC2 leftHI;

3Ris leftNC2 leftHI;

4Ris left min-AP-injective leftHI;

5Ris left idempotent reflexive leftHI.

Acknowlegment

Project supported by the Foundation of Natural Science of China10771182, 10771183.

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