International Journal of Mathematics and Mathematical SciencesVolume 2011 (2011), Article ID 154636, 10 pagesdoi:10.1155/2011/154636

Review Article

Some Notes on Semiabelian Rings

Junchao Wei and Nanjie Li

School of Mathematics, Yangzhou University, Yangzhou 225002, China

Received 12 January 2011; Accepted 6 June 2011

Academic Editor: Frank Sommen

Copyright © 2011 Junchao Wei and Nanjie Li. 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.

It is proved that if a ring is semiabelian, then so is the skew polynomial ring , where is an endomorphism of satisfying for all . Some characterizations and properties of semiabelian rings are studied.

1. Introduction

Throughout the paper, all rings are associative with identities. We always use and to denote the set of all nilpotent elements and the set of all idempotent elements of .

According to [1], a ring is called semiabelian if every idempotent of is either right semicentral or left semicentral. Clearly, a ring is semiabelian if and only if either or for every , so, abelian rings (i.e., every idempotent of is central) are semiabelian. But the converse is not true by [1, Example 2.2].

A ring is called directly finite if implies for any . It is well known that abelian rings are directly finite. In Theorem 2.7, we show that semiabelian rings are directly finite.

An element of a ring is called a left minimal idempotent if and is a minimal left ideal of . A ring is called left min-abel [2] if every left minimal idempotent element of is left semicentral. Clearly, abelian rings are left min-abel. In Theorem 2.7, we show that semiabelian rings are left min-abel.

A ring is called left idempotent reflexive if for any and , implies . Theorem 2.5 shows that is abelian if and only if is semiabelian and left idempotent reflexive.

In [3], Wang called an element of a ring an op-idempotent if . Clearly, op-idempotent need not be idempotent. For example, let . Then is an op-idempotent, while it is not an idempotent. In [4], Chen called an element  potent in case there exists some integer such that . We write for the smallest number of such. Clearly, idempotent is potent, while there exists a potent element which is not idempotent. For example, is a potent element, while it is not idempotent. We use and to denote the set of all op-idempotent elements and the set of all potent elements of . In Corollaries 2.2 and 2.3, we observe that every semiabelian ring can be characterized by its op-idempotent and potent elements.

If is a ring and is a ring endomorphism, let denote the ring of skew polynomials over ; that is all formal polynomials in with coefficients from with multiplication defined by . In [1], Chen showed that is a semiabelian ring if and only if is a semiabelian ring. In Theorem 2.13, we show that if is a semiabelian ring with an endomorphism satisfying for all , then is semiabelian.

2. Main Results

It is well known that an idempotent of a ring is left semicentral if and only if is right semicentral. Hence we have the following theorem.

Theorem 2.1. The following conditions are equivalent for a ring . (1) is a semiabelian ring.(2)For any , is an ideal of . (3)For any , .

Proof. (1)⇒(2) assume that . Since is semiabelian, is either left semicentral or right semicentral. If is right semicentral, then and is left semicentral. Thus and is an ideal of . Similarly, if is left semicentral, then is also an ideal of .(2)⇒(3) is clear.(3)⇒(1) assume that . If is neither left semicentral nor right semicentral, there exist such that and . By , . If , then , a contradiction; if , then , it is also a contradiction. Hence is either left semicentral or right semicentral.

Evidently, is semiabelian if and only if either or for every . On the other hand, an element of is op-idempotent if and only if is idempotent. Hence, by Theorem 2.1, we have the following corollary.

Corollary 2.2. The following conditions are equivalent for a ring . (1) is a semiabelian ring.(2)For any , or .(3)For any , is an ideal of .(4)For any , .

Clearly, for any , , , and . Hence, by Theorem 2.1, we have the following corollary.

Corollary 2.3. The following conditions are equivalent for a ring . (1) is a semiabelian ring.(2)For any , or .(3)For any , is an ideal of .(4)For any , .

Using Theorem 2.1, Corollaries 2.2 and 2.3, we have the following corollary.

Corollary 2.4. Let be a semiabelian ring. If , and , then: (1)if , then ,(2)if , then ,(3)if , then .

Call a ring   idempotent reversible if implies for . Clearly, abelian rings are left idempotent reflexive, and left idempotent reflexive rings are idempotent reversible. But we do not know that whether idempotent reversible rings must be left idempotent reflexive. It is easy to see that a ring is left idempotent reflexive if and only if for any , implies . (In fact, it is only to show the sufficiency: Let and satisfy . If , then for some . Since and , by hypothesis, , this implies , which is a contradiction. Hence , is a left idempotent reflexive ring.)

Let be a division ring. Then the 2-by-2 upper triangular matrix ring is not idempotent reversible. In fact, and , but . On the other hand, by [1, Example 2.2], is a semiabelian ring.

We have the following theorem.

Theorem 2.5. The following conditions are equivalent for a ring . (1) is an abelian ring.(2) is a semiabelian ring and idempotent reversible ring.(3) is a semiabelian ring and left idempotent reflexive ring.(4) is a semiabelian ring and for any , implies .

Proof. (1)⇒(3)⇒(2)⇒(1) and (3)⇒(4) are trivial.Now let . If is semiabelian, then is either right semicentral or left semicentral. If is right semicentral, then . Since , (4) implies . Hence . This shows that is central; if is left semicentral, then is right semicentral. Hence and so is also central. Thus (4)⇒(1) holds.

Since semiprime rings are left idempotent reflexive, we have the following corollary by Theorem 2.5.

Corollary 2.6. Semiprime semiabelian rings are abelian.

Theorem 2.7. Let be a semiabelian ring and . Then, (1),(2)If and , then for all ,(3).

Proof. (1) Since is right semicentral if and only if and is left semicentral if and only if , is evident by hypothesis.(2)Since , by (1). Hence , so for any , .(3)Since , by (2), for all . This implies for all . Hence .

Theorem 2.8. Let be a semiabelian ring. Then, (1) is directly finite,(2) is left min-abel.

Proof. (1) Assume that . Let . Then and . By Theorem 2.7(3), . Hence , which implies .(2) Let and be a minimal left ideal of . Then and . Since is a semiabelian ring, by Theorem 2.7(3), . This implies , that is, which is a contradiction. Hence , so is left semicentral. Hence is a left min-abel ring.

For a ring , a proper left ideal of is prime if implies that or . Let be the set of all prime left ideals of . In [5], it has been shown that if is not a left quasiduo ring, then is a space with the weakly Zariski topology but not with the Zariski topology.

Let be a ring. Then the set of all maximal left ideals of is a compact -space by [6, Lemma 2.1]. Recall that a topological space is said to be zero dimensional if it has a base consisting of clopen sets. Where a clopen set in a topological space is a set which is both open and closed.

Now, for a left ideal of a ring , let and . If for some , then we write and for and .

For any left ideal of , we let , and let .

A ring is called left topologically boolean, or a left -ring [7] for short, if for every pair of distinct maximal left ideals of there is an idempotent in exactly one of them.

A ring is called clean [8] if every element of is the sum of a unit and an idempotent.

The following theorems generalize [6, Lemmas 2.2 and 2.3].

Lemma 2.9. Let be a semiabelian ring and . Then, (1)if is a maximal left ideal of and , then ,(2),(3),(4),(5),(6),(7) .In particular, every set in is clopen.

Proof. (1) Since , . Let for some and . Since by Theorem 2.7(1), . Since is a prime left ideal and , .(2) Let . Then and . By (1), we have . Hence . Clearly, , so . This shows . Conversely, if , then . Since is a left ideal, . Hence by (1). If , then implies , which is a contradiction. Hence , so . Therefore . Thus . Similarly, we have .(3) and (4) They are also straightforward to prove.By induction on , we can show (5), (6) and (7).Thus every set in is clopen.

Theorem 2.10. Let be a semiabelian clean ring. Then is a left -ring.

Proof. Suppose that and are distinct maximal left ideals of . Let . Then and for some . Clearly, . Since is clean, there exist an idempotent and a unit in such that . If , then from which it follows that , a contradiction. Thus . If , then by Lemma 2.9 (1) and hence . It follows that which is also not possible. We thus have that is an idempotent belonging to only.

Theorem 2.11. Let be a semiabelian ring. If is a left -ring, then forms a base for the weakly Zariski topology on . In particular, is a compact, zero-dimensional Hausdorff space.

Proof. Similar to the proof of [6, Proposition 2.5], we can complete the proof.

A ring is called von Neumann regular if for all and is said to be unit-regular if for any , for some . A ring is called strongly regular if for all . Clearly, strongly regular unit-regular von Neumann regular. Since von Neumann regular rings are semiprime, it follows that von Neumann regular rings are left idempotent reflexive. And it is well known that is strongly regular if and only if is von Neumann regular and abelian. In view of Theorem 2.5, we have the following corollary.

Corollary 2.12. The following conditions are equivalent for a ring . (1) is strongly regular.(2) is unit-regular and semiabelian.(3) is von Neumann regular and semiabelian.

Following [9], a ring is called left if for any , is projective left -module, and is said to be -regular if for any , . A ring is said to be reduced if implies for each , or equivalently, . Obviously, reduced rings are -regular and abelian, and -regular rings are left and semiprime. Using Theorem 2.5, the following theorem gives some new characterization of reduced rings in terms of semiabelian rings.

Theorem 2.13. The following conditions are equivalent for a ring . (1) is reduced.(2) is -regular and semiabelian.(3) is left , semiprime, and semiabelian.

Proof. (1)⇒(2)⇒(3) are trivial.(3)⇒(1) let such that . Since is left , . Hence and because . Since is semiabelian and , by Theorem 2.7. Since is semiprime, , which shows that is reduced.

If is a ring and is a ring endomorphism, let denote the ring of skew polynomials over ; that is all formal polynomials in with coefficients from with multiplication defined by . Note that if is the -bimodule defined by and , for all and , then .

Theorem 2.14. Let be a semiabelian ring. If is a ring endomorphism of satisfying for all . Then is semiabelian.

Proof. Let . Then Since , by hypothesis. Hence we have the following equations: If is right semicentral, then , which implies . Hence .Assume that and for . Then so Hence and .For any , we have . Thus , which implies is right semicentral in . Similarly, if is left semicentral in , then we can show that is left semicentral in . Hence is a semiabelian ring.

Corollary 2.15. Let be a semiabelian ring. If is a ring endomorphism of satisfying for all . Then is semiabelian.

Proof. Since every element of can be written with , by the same proof as Theorem 2.14, we can complete the proof.

Corollary 2.16. Let be a semiabelian ring. If is a ring endomorphism of satisfying for all . Then is semiabelian.

Corollary 2.17. Let be a division ring with an endomorphism , then is semiabelian.

A ring is called left [2] if for any and , implies . Clearly, left idempotent reflexive rings are left . We do not know whether idempotent reversible rings are left . But we know that there exists a left ring which is not idempotent reversible. In fact, there exists a semiabelian ring which is not abelian (see the example above Theorem 2.5), by [1, Corollary 2.4], is a semiabelian ring which is not abelian. Hence, by Theorem 2.5, is not idempotent reversible. But is a left ring.

The authors in [10, Theorem 4.1] showed that if is a left ring containing an injective maximal left ideal, then is a left self-injective ring. And [11, Proposition 5] showed that if is a left idempotent reflexive ring containing an injective maximal left ideal, then is a left self-injective ring.

Proposition 2.18. Let be an idempotent reversible 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 we have . Since is idempotent reversible, . So is central. Now let be any proper essential left ideal of and any non-zero 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 [10, Proposition 5], we have that is injective. Hence is left self-injective.

Recall that a ring is left if every principal left ideal of is projective. As an application of Proposition 2.18, we have the following result.

Corollary 2.19. The following conditions are equivalent for a ring . (1) is a von Neumann regular left self-injective ring with .(2) is an idempotent reversible left ring containing an injective maximal left ideal.

Proof. (1)⇒(2) is trivial.(2)⇒(1) by Proposition 2.18, is a left self-injective ring. Hence, by [12, Theorem 1.2], is left , so, is von Neumann regular because is left . Also we have since there is an injective maximal left ideal.

By [13], a ring is said to be left if is left hereditary containing an injective maximal left ideal. Osofsky [14] proved that left self-injective left hereditary ring is semisimple Artinian. We can generalize the result as follows.

Corollary 2.20. The following conditions are equivalent for a ring . (1) is a semisimple Artinian ring.(2) is an idempotent reversible left ring.

According to [8], an element is called exchange if there exists such that and , and is said to be clean if where and . By [8], clean elements are exchange and the converse holds when is an abelian ring. A ring is called exchange (clean) ring if every element of is an exchange (clean) element.

Proposition 2.21. Let be a semiabelian ring. If is an exchange element, then is a clean element.

Proof. Since is an exchange element, there exists such that and . Let and where . Then . Since is a semiabelian ring, is either left semicentral or right semicentral. If is left semicentral, then and by Theorem 2.7(1). Hence . Similarly, if is right semicentral, then and . This implies , so . Since is a semiabelian ring, by Theorem 2.8, is a directly finite ring. Hence , which implies is a clean element.

Corollary 2.22. If is a semiabelian exchange ring, then is a clean ring.

Theorem 2.23. Let be a semiabelian ring and . If , then .

Proof. Let and . Since is a semiabelian ring, either is left semicentral or is right semicentral. If is left semicentral, then . If is right semicentral, then . Hence for each , which implies .

Corollary 2.24. Let be an abelian ring and . If , then .

The converse of Corollary 2.24 is not true, in general.

Example 2.25. let be a field, and . Evidently,  ,, . Let and . Then . Since and . Hence , but is not an abelian ring.

A ring is called EIFP if , implies . Clearly, semiabelian rings are EIFP by Theorem 2.23. But the converse of Theorem 2.23 is not true, in general.

Example 2.26. Take the ring in Example 2.25, and let . Then is EIFP, but not semiabelian. Indeed, take and in , where are matrix units. Then is an idempotent. By a direct computation, is neither left nor right semicentral. Hence is not semiabelian while is EIFP.

Acknowledgments

This paper was supported by the Foundation of Natural Science of China (10771182) and the Scientific Research Foundation of Graduate School of Jiangsu Province (CX09B_309Z). The authors would like to thank the referee for his/her helpful suggestions and comments.

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