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International Journal of Mathematics and Mathematical Sciences
Volume 2011 (2011), Article ID 294301, 8 pages
http://dx.doi.org/10.1155/2011/294301
Research Article

Left Rings

School of Mathematics, Yangzhou University, Yangzhou 225002, China

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

Academic Editor: Frank Werner

Copyright © 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.

Abstract

We introduce in this paper the concept of left rings and concern ourselves with rings containing an injective maximal left ideal. Some known results for left idempotent reflexive rings and left rings can be extended to left 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, denotes an associative ring with identity, and all modules are unitary. For any nonempty subset of a ring , and denote the set of right annihilators of and the set of left annihilators of , respectively. We use , , , , , , and 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 , respectively.

An element of is called left minimal if is a minimal left ideal. An element of is called left minimal idempotent if is left minimal. We use and for the set of all left minimal elements and the set of all left minimal idempotent elements of , respectively. Moreover, let .

A ring is called left if every minimal left ideal which is isomorphic to a summand of is a summand. Left rings were initiated by Nicholson and Yousif in [1]. In [26], the authors discussed the properties of left rings. In [1], a ring is called left mininjective if for every , and is said to be left minsymmetric if always implies . According to [1], left mininjective left minsymmetric left , and no reversal holds.

A ring is called left universally mininjective [1] if is an idempotent left ideal of for every . The work in [2] uses the term left for the left universally mininjective. According to [1, Lemma  5.1], left rings are left mininjective.

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

A ring is called left if implies for and .

Let be a field and . Then and is empty, so is left . Now let . Then and . Since and , is not left .

Let be any ring and and . Then and are all empties, so and are all left .

A ring is called left idempotent reflexive [8] if implies for all and . Clearly, is left idempotent reflexive if and only if for any and , implies if and only if for any and , implies . Therefore, left idempotent reflexive rings are left .

In general, the existence of an injective maximal left ideal in a ring can not guarantee the left self-injectivity of . In [9], Osofsky proves that if is a semiprime ring containing an injective maximal left ideal, then is left self-injective. In [8], Kim and Baik prove that if is left idempotent reflexive containing an injective maximal left ideal, then is left self-injective. In [10], Wei and Li prove that if is left containing an injective maximal left ideal, then is left self-injective. Motivated by these results, in this paper, we show that if is a left ring containing an injective maximal left ideal, then is left self-injective. As an application of this result, we show that a ring is a semisimple Artinian ring if and only if is a left ring and left ring.

We start with the following theorem.

Theorem 1. The following conditions are equivalent for a ring : (1) is left ;(2)for any and , implies ;(3)for any , implies that ;(4)for any , implies .

Proof. (1) (2) Assume that and with . If , then . By (1), for some . Hence , which is a contradiction. Hence .
(2) (3) Let such that . Then . By (2), . Hence .
(3) (4) Assume that with . If , then . Hence , which implies for some . Since , there exists such that . Let . Then because and . Since , by (3), . Hence , which implies . It is a contradiction. Therefore .
(4) (1) Let and with . Then there exists such that and . If , then , so , by (4), , which implies . It is a contradiction. Hence , so for some , which implies is a left ring.

Corollary 2. Left rings are left .

Proof. Let and with . If , then for some . Clearly, and . Since is a left ring, by Theorem 1, , which implies , and this is a contradiction. Hence and so is a left 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 be a ring. Then the following conditions are equivalent: (1) is a left ring;(2)for any and , implies ;(3)for any , ;(4)for any and , implies .

Proof. (4) (1) (2) It is easy to show by the definition of left ring.
(2) (3) Let . Then . We claim that . Otherwise, there exists such that . Clearly, and . By (2), we have . But because . This is a contradiction. Hence and so . Therefore .
(3) (4) Since , is projective. It is easy to show that and for some . Since , . Therefore , which implies . By (3), . Hence .

By Theorem 3, we have the following corollary.

Corollary 4. (1) Let be a left ring. If satisfying , then is left .
(2) If is a direct product of a family rings , then is a left ring if and only if every is left .

Theorem 5. (1) If is a subdirect product of a family left rings , then is a left ring.
(2) If is a left ring, so is .

Proof. (1) Let , where are ideals of with . Let and satisfying . For any , if , then ; if , then we can easily show that . Since is a left ring and , where , . Hence . In any case, we have for all . Therefore and so . This shows that is a left ring.
(2) Let and satisfying . Clearly, in , , . Since , . Since is a left ring, by Theorem 3, , which implies . Thus is a left ring by Theorem 3.

Theorem 6. (1)   is a strongly left min-abel ring if and only if is a left min-abel left ring.
(2) If is a strongly left min-abel ring, then so is .

Proof. (1) Theorem  1.8 in [3] shows that is a strongly left min-abel ring if and only if is a left min-abel left ring, so by Corollary 2, we obtain that strongly left min-abel ring is left min-abel left .
Conversely, let be a left min-abel left ring. Let and satisfying . Set . Then, clearly, and . Since is a left min-abel ring, , so . Since is a left ring, , which implies , by Theorem 1, is a left ring. Hence 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 is called left idempotent reflexive [8] if implies for all and . Clearly, left idempotent reflexive rings are left .

In general, the existence of an injective maximal left ideal in a ring cannot guarantee the left self-injectivity of . Proposition  5 in [8] proves that if is a left idempotent reflexive ring containing an injective maximal left ideal, then is a left self-injective ring. Theorem  4.1 in [10] proves that if is a left MC2 ring containing an injective maximal left ideal, then is a left self-injective ring. We can generalize the results as follows.

Theorem 7. Let be a left ring. If contains an injective maximal left ideal, then is a left self-injective ring.

Proof. Let be an injective maximal left ideal of . Then for some minimal left ideal of . Hence we have and for some . If , then . Since is left and , . So is central. Now let be any proper essential left ideal of and any nonzero left homomorphism. Then , where is a maximal submodule of . Now , where is a minimal left ideal of . Since is central, . For any , let , where , . Then . Since , . Since , . Thus . Hence is injective. If , by the proof of [11, Proposition  5], we have is injective. Hence is left self-injective.

A ring is called strongly left [3] if for all . Since strongly left left left mininjective left minsymmetric left left and strongly left min-abel left , we have the following corollary.

Corollary 8. Let contain an injective maximal left ideal. If satisfies one of the following conditions, then is a left self-injective ring. (1) is a strongly left ring.(2) is a left ring.(3) is a left mininjective ring.(4) is a left minsymmetric ring.(5) is a strongly left min-abel ring.(6) is a left ring.

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

Corollary 9. Let contain an injective maximal left ideal. Then is left self-injective if and only if .

A ring is called left injective [5] if for any , , and is said to be left [5] if for any , is projective implies that for some . By [5, Theorem  2.22], left injective rings are left and left rings are left . A ring is right Kasch if every simple right module can be embedded in , and is said to be left [12] if every left ideal that is isomorphic to a direct summand of is itself a direct summand. Clearly, left self-injective rings are left [13] and left rings are left and by [14, Lemma  1.15], right Kasch rings are left . Hence, we have the following corollary.

Corollary 10. (1) Let contain an injective maximal left ideal. Then the following conditions are equivalent: (a) is a left self-injective ring;(b) is a left nil-injective ring;(c) is a left ring;(d) is a left ring.
(2) If is a right Kasch ring containing an injective maximal left ideal, then is a left self-injective ring.

A ring is called left min- injective if for any , , where is a right ideal of . Clearly, left mininjective rings are left min injective.

Lemma 11. (1) If is a left min injective ring, then is left .
(2) If , then is left .

Proof. (1) Let and satisfying . Since is a left min injective ring and , , where is a right ideal . Set , and . Then , so , which implies , so . Let . Then and . Therefore which implies . By Theorem 3, is a left ring.
(2) Assume that and satisfying . If , then . Thus there exists a minimal right ideal of such that . Clearly, and . Hence . Let be a complement right ideal of in . Then and , which is a contradiction. Hence . By Theorem 1, is a left ring, so is left by Corollary 2.

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

Theorem 12. Let contain an injective maximal left ideal. Then the following conditions are equivalent: (1) is left self-injective;(2) is left min injective;(3) .

A ring is called(1)strongly reflexive if implies for all ;(2)reflexive [8, 15] if implies for all ;(3)symmetric if implies for all ;(4) [16] if implies for all ;(5) [16] if implies for all .

Evidently, we have the following proposition.

Proposition 13. (1) The following conditions are equivalent for a ring : (a) is semiprime;(b) is strongly reflexive and every proper essential right ideal of contains no nonzero nilpotent ideal;(c) is reflexive and every proper essential right ideal of contains no nonzero nilpotent ideal;(d) is strongly reflexive and ;(e) is reflexive and .
(2) is symmetric if and only if is and strongly reflexive.
(3) is reversible if and only if is and reflexive.
(4) Strongly reflexive reflexive left idempotent reflexive.

It is well known that if is a left self-injective ring, then , so is semiprimitive. Hence is left by Proposition 13. Thus, by Theorems 5 and 7, we have the following theorem.

Theorem 14. Let contain an injective maximal left ideal. Then (1) is a left self-injective ring if and only if is a left ring.(2)If satisfies one of the following conditions, then is a left self-injective:(a) is a semiprime ring;(b) is a strongly reflexive;(c) is reflexive;(d) is a left idempotent reflexive.

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

Theorem 15. The following conditions are equivalent for a ring : (1) is a von Neumann regular left self-injective ring with ;(2) is a left left ring containing an injective maximal left ideal;(3) is a left ring containing an injective maximal left ideal and is a left ring.

Proof. (1) (3) is trivial.
(3) (2) is a direct result of Theorem 5(2).
(2) (1) By Theorem 7, is left self-injective. Hence, by [13, Theorem  1.2], is left , so is von Neumann regular because is left . In addition, we have since there is an injective maximal left ideal.

By [17], a ring is said to be left if 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 : (1) is semisimple Artinian;(2) is left left ;(3) is left left ;(4) is left min- injective left ;(5) is left idempotent reflexive left .

Acknowlegment

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

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