Research Article | Open Access
Areej M. Abduldaim, Sheng Chen, "The Unique Maximal GF-Regular Submodule of a Module", The Scientific World Journal, vol. 2013, Article ID 750808, 3 pages, 2013. https://doi.org/10.1155/2013/750808
The Unique Maximal GF-Regular Submodule of a Module
An -module is called -regular if, for each and , there exist and a positive integer such that . We proved that each unitary -module contains a unique maximal -regular submodule, which we denoted by . Furthermore, the radical properties of are investigated; we proved that if is an -module and is a submodule of , then . Moreover, if is projective, then is a -pure submodule of and .
Throughout this paper, is a commutative ring with identity and all modules are left unitary, unless otherwise stated. Recall that an element is said to be regular if there exists such that ; a ring is called regular if and only if each element of is regular. An ideal of a ring is regular if each of its elements is regular in ; indeed, a regular ideal of is itself a regular ring . Brown and McCoy proved in  that each ring contains a unique maximal regular ideal which satisfies the well-known radical properties. The ideal is called the regular radical of .
The concept of regularity was extended to modules in several ways and in  the notion of -regular modules (in the sense of Fieldhouse ) was generalized to -regular modules. Let be an -module; an element is said to be -regular if for each there exist and a positive integer such that . An -module is called -regular if and only if all its elements are -regular; in particular, a ring is -regular if and only if is -regular as an -module. On the other hand a ring is -regular if and only if is a -regular -module; recall that a ring is -regular if, for each , there exist and a positive integer such that . A submodule of an -module is called -regular if each element of is -regular and every submodule of a -regular module is a -regular module. Also, in  the concept of -pure submodules was introduced; a submodule of an -module is called -pure if, for each , there exists a positive integer such that .
In this paper we show that each module contains a unique maximal -regular submodule, which we denote by , and we show that satisfies some but not all of the usual radical properties.
2. Main Results
Theorem 1. Let be any ring. Every -module contains a unique maximal -regular submodule.
Proof. Let be any ring, let be an -module, and let where because is a -regular submodule of . Let be an ascending chain in and . Let ; there exists such that , but is a -regular submodule; then; for each , there exist and a positive integer such that ; therefore is a -regular element in which implies that is a -regular -module. Now, by Zorn’s lemma, contains a maximal element which we call . To prove the uniqueness of , assume that and be two maximal -regular submodules in ; then for any maximal ideal of each of and is semisimple over [2, Proposition 21]. Now, let ; then and ; thus and , where and are two submodules of . Hence, , but each of , , and is a semisimple submodule; thus is a semisimple submodule which implies that is -regular . So is a -regular submodule [2, Theorem 20]. Now, each of and is a maximal -regular submodule and hence .
Remark 2. We denote the unique maximal -regular submodule of an -module by . It is obvious that contains every -regular submodule of ; this means that is a -regular submodule which is not contained properly in any other -regular submodule. In fact, is the sum of all -regular submodules of and if and only if is a -regular module.
Example 3. (a) Since the -module is -regular for each positive integer , then .
(b) Each element in the -module is not -regular ; hence .
(c) Let be a prime number and let be a -module. Let ; then there exists a positive integer such that , but is a -regular -module for each positive integer ; hence is a -regular element, so which implies that .
(d) Let be a -module; since is a torsion -module, then is a -regular -module [2, Proposition 6]. Since for each prime number , then for all primes .
Proposition 4. Let and be -modules, and let be a submodule of ; then (a), (b).
Proof. (a) Let be a submodule of , and let ; then and is -regular in which implies that is -regular in , thus . Conversely, let and ; therefore is -regular in which means that and hence .
(b) Let ; then , where and . Since is -regular, then each of and is -regular which means that and ; hence .
Proposition 5. Let and be -modules, and let be an -homomorphism; then .
Proof. If , then is a -regular ring, but ; thus is an epimorphic image of ; hence it is a -regular ring. Therefore, and .
Remark 6. (a) If is an -epimorphism, then in general. In fact, let be the natural map, where and are -modules. It is easy to check that , , but .
(b) It is shown in  that, for a ring , which is not true in case of -regular modules; this means that (as in (a)).
Corollary 7. For each -module , .
Proof. For each , let be an -homomorphism defined by . Then by Proposition 5, but ; hence .
Let be the Jacobson radical of a ring . Brown and McCoy proved in  that . However, this is not true for -regular modules; for example, if is a -module, then , and .
Lemma 8. Let be an -module and let be a -pure submodule of . For any , there exists a positive integer such that if and only if .
Proof. Since is -pure in , then, for each , there exists a positive integer such that . If , then , and hence . Conversely, if , then , but ; therefore .
Lemma 9. Let ; if is a finitely generated -pure submodule of an -module such that for some positive integer , then .
Theorem 10. Let be an -module. If is a -pure submodule of , then .
Recall that is always a pure ideal in . Hence is -pure .
Theorem 11. Let be a projective -module; then (a), (b) is a -pure submodule of , (c).
Proof. (a) By the dual basis lemma , for each we have that , where , for all and . If , then the submodule is -regular and is a -regular ideal in by Proposition 5, hence . Thus . We get the other direction of the inclusion by Corollary 7.
(b) First we claim that; for any two ideals and of , ; it is enough to show this locally; thus we may assume that is free. It is clear that . On the other hand, let ; then By freeness, and . Now, let be any ideal in ; then by (a) we get . But is -pure ideal, so, for each , there exists a positive integer such that ; hence .
(c) Since is projective, then  which implies that by Theorem 10.
Corollary 12. Let be any ring, and let be any -module such that is a -pure submodule and ; then .
Proof. Since , then by Theorem 10 we get that .
Remark 13. If is a -regular -module, then ; hence . In fact, this shows that Theorem 10 is a generalization of [2, Proposition 28].
In  we noticed that every module over -regular ring is -regular, but the converse need not be true in general. The next result shows how the converse may be true, but first we recall that if is an -module, then the trace of is , where .
Proposition 14. Let be a -regular -module. If , then is a -regular ring.
Proof. For each and , since is a -regular submodule of , then by [2, Proposition 7] we get that is a -regular ideal. Thus , but ; hence , which implies that and is -regular.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Sheng Chen was supported by the National Natural Science Foundation of China (Grants nos. 11001064 and 11101105), by the Fundamental Research Funds for the Central Universities (Grant no. HIT. NSRIF. 2014085), and by the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry, and Areej M. Abduldaim was supported by China Scholarship Council (CSC no. 2011368T09).
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Copyright © 2013 Areej M. Abduldaim and Sheng Chen. 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.