ISRN Algebra

Volume 2014 (2014), Article ID 603851, 4 pages

http://dx.doi.org/10.1155/2014/603851

Research Article

Generalized -Radical Supplemented Modules

Burcu Nişancı Türkmen¹ and Ali Pancar²

¹Faculty of Art and Science, Amasya University, İpekköy, 05100 Amasya, Turkey

²Department of Mathematics, Faculty of Art and Science, Ondokuz Mayis University, Kurupelit, 55139 Samsun, Turkey

Received 28 October 2013; Accepted 16 December 2013; Published 14 January 2014

Academic Editors: A. V. Kelarev and S. Yang

Copyright © 2014 Burcu Nişancı Türkmen and Ali Pancar. 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.

Çalışıcı and Türkmen called a module generalized -supplemented if every submodule has a generalized supplement that is a direct summand of . Motivated by this, it is natural to introduce another notion that we called generalized -radical supplemented modules as a proper generalization of generalized -supplemented modules. In this paper, we obtain various properties of generalized -radical supplemented modules. We show that the class of generalized -radical supplemented modules is closed under finite direct sums. We attain that over a Dedekind domain a module is generalized -radical supplemented if and only if is generalized -radical supplemented. We completely determine the structure of these modules over left -rings. Moreover, we characterize semiperfect rings via generalized -radical supplemented modules.

1. Introduction

Throughout the whole text, all rings are to be associative; unit and all modules are left unitary. We specially mention [1–4] among books concerning the structures of modules and rings. We shall write () if is a submodule of (small in ). By , we denote the radical of . Let . is called a supplement of in if it is minimal with respect to . is a supplement of in if and only if and [4]. A module is called supplemented if every submodule of has a supplement, and it is called -supplemented if every submodule of has a supplement that is a direct summand of [5]. Clearly every -supplemented module is supplemented. In [6], Zöschinger introduced a notion of modules whose radical has supplements called radical supplemented. Xue defined generalized supplemented modules as another generalization of supplemented modules [7]. Let be any submodule of . If there exists a submodule of such that and , is called a generalized supplement of in . is called generalized supplemented if every submodule of has a generalized supplement in . Also Çalışıcı and Türkmen called a module   generalized -supplemented if every submodule has a generalized supplement that is a direct summand of as a generalization of -supplemented modules [8]. So it is natural to introduce another notion that we called generalized -radical supplemented modules. A module is called generalized -radical supplemented if every submodule containing radical has a generalized supplement that is a direct summand of .

In this paper we obtain various properties of generalized -radical supplemented modules as a proper generalization of generalized -supplemented modules. We prove the following indications.

(i)Every generalized -radical supplemented module has a radical direct summand.(ii) is a generalized -radical supplemented module for every -module .(iii)The class of generalized -radical supplemented modules is closed under finite direct sums.(iv)If is a generalized -radical supplemented module, then is a generalized -radical supplemented module for every fully invariant submodule of .(v)Let be any infinite collection of generalized -radical supplemented modules, and let . If is a duo module, then is generalized -radical supplemented.(vi)Over a Dedekind domain a module is generalized -radical supplemented if and only if is generalized -radical supplemented.(vii)Every generalized -radical supplemented -module is injective if and only if is left Noetherian -ring.(viii)A ring is semiperfect if and only if every finitely generated free -module is generalized -radical supplemented.

2. Generalized -Radical Supplemented Modules

Definition 1. A module is called generalized -radical supplemented if every submodule containing radical has a generalized supplement that is a direct summand of .

Recall that a module is called radical if has no maximal submodules; that is, . For a module , will indicate the sum of all radical submodules of . If , is called reduced. Note that is the largest radical submodule of [4].

Now we have the following simple fact, which lays a key role in our study.

Lemma 2. Let be an -module. If , then is a generalized -radical supplemented module.

Proof. Since , has the trivial generalized supplement in . Consequently, is a generalized -radical supplemented module.

Corollary 3. is a generalized -radical supplemented module for every -module .

Proposition 4. Every generalized -radical supplemented module has a radical direct summand.

Proof. Let be a generalized -radical supplemented module. Then there exist submodules and of such that , , and . Since by [4, Theorem 21.6], then . Applying the modular law, we obtain that .

Recall from [4] that a submodule of an -module is called fully invariant if is contained in for every -endomorphism of . A module is called duo, if every submodule of is fully invariant [9].

Theorem 5. The following statements hold over a ring .

(1)Let be any finite collection of modules and . Then is generalized -radical supplemented if, for each , is generalized -radical supplemented.(2)Let be any infinite collection of modules and . If is a duo module, then is generalized -radical supplemented if, for each , is generalized -radical supplemented.

Proof. Let be a generalized -radical supplemented module for each . To prove that is a generalized -radical supplemented module it is sufficient by induction on to prove this is the case when . Hence, suppose that . Let be any submodule of with . Then so that has a generalized supplement in . Since , then . It follows that has a generalized supplement in such that is a direct summand of . By [10, Proposition 2.4], is a generalized supplement of in . Moreover, . Since is a generalized -radical supplemented module, has a generalized supplement in such that is a direct summand of . Again applying [10, Proposition 2.4], is a generalized supplement of in . It is clear that is a direct summand of . Therefore, is a generalized -radical supplemented module.

Let be any submodule such that . Then for every . By this hypothesis, there exist submodules and of such that , , and for every . Since is a fully invariant submodule of , then . Let and . Then there exist submodules and of such that , by [4, Theorem 21.6] and . Thus, is a generalized -radical supplemented module.

Proposition 6. If is a generalized -radical supplemented module, then is a generalized -radical supplemented module for every fully invariant submodule of .

Proof. Let be any fully invariant submodule of and let be any submodule of with . Since , we have . By the hypothesis, we have , , and for some submodules and of . By [11, Lemma 2.3] is generalized supplement of in . Since is a fully invariant submodule of , we have and . Then . Hence, is a generalized -radical supplemented module.

Corollary 7. Let be a Dedekind domain and let be an -module. is a generalized -radical supplemented module if and only if is a generalized -radical supplemented module.

Proof. We know that is a fully invariant submodule of . So, by Proposition 6, is a generalized -radical supplemented module. Conversely, suppose that is a generalized -radical supplemented module. Since is a Dedekind domain, we have for some submodule of . By the hypothesis, is a generalized -radical supplemented module. Hence, is a generalized -radical supplemented module by Theorem 5(1) and Corollary 3.

Example 8. Consider the left -module . According to Lemma 2, is a generalized -radical supplemented module. On the other hand, is not -supplemented because it is not torsion.

Theorem 9. Let be a module. Suppose that is small in . Then is a generalized -radical supplemented module if and only if it is -supplemented.

Proof. Let be any submodule of . Then . Since is a generalized -radical supplemented module, we have , , and for submodules . Since , we have , . Therefore is -supplemented.

is trivial.

Corollary 10. Every generalized -radical supplemented module over a left Bass ring is -supplemented.

Proposition 11. Every direct summand of is a generalized -radical supplemented module for every -module .

Proof. Let be any direct summand of . Then there exists a submodule of such that . Since by [4, Theorem 21.6], then . By Lemma 2, is generalized -radical supplemented.

Proposition 12. Let be a generalized -radical supplemented module. Suppose that a cofinite fully invariant submodule of is a direct summand of . Then is generalized -radical supplemented.

Proof. Let be any submodule of with . By the hypothesis, we have for some finitely generated submodule of . Then . Clearly . So there exist submodules of such that , , and . Since , we have , and . It follows that and . Since is a fully invariant submodule of , then . Note that . Therefore, is generalized -radical supplemented.

Lemma 13. Let be a module and a submodule such that . If is a direct summand of , then . In particular, if is a direct summand of , then .

Proof. Let be any direct summand of such that . Then there exists a submodule of such that . By [4, Theorem 21.6], . Since , . Now we take under the similar condition. ; that is, is radical. Consequently, .

Let be a ring and let be an -module. We consider the following condition.

If and are direct summands of with , then is also a direct summand of .

Proposition 14. Let be a generalized -radical supplemented module with and let be a submodule with . If is a direct summand of , is generalized -radical supplemented.

Proof. Let be a submodule of such that . By Lemma 13, . Since is a generalized -radical supplemented module, there exist submodules of such that , , and . Then . Since satisfies , is a direct summand of . Then there exists a submodule of such that . It follows that and . Therefore is a generalized -radical supplemented module.

A module is called local if is simple and is small in [10]. The module is called -local if is simple [12].

The following fact is clear.

Corollary 15. Every -local module is generalized -radical supplemented.

Lemma 16. Let be an indecomposable module. If is a generalized -radical supplemented module, then or is -local.

Proof. Suppose that . Then contains at least one maximal submodule, say . Since , then there exist submodules of such that , , and . We have because is indecomposable. Since is a generalized supplement of , is -local by [12, Lemma 3.3]. So is -local.

Theorem 17. Let be a local commutative ring and be a uniform -module. Every submodule of is generalized -radical supplemented if and only if it is uniserial.

Proof. By [13, Lemma 6.2], it suffices to show that every finitely generated submodule of is local. Let be any finitely generated submodule of . By assumption, is indecomposable. So, by Lemma 16, is -local. Since is finitely generated, is local.

Since is uniserial, every submodule of is hollow by [3, 2.17]. So it is easy to see that every submodule of is a generalized -radical supplemented module.

Corollary 18. Let be a local commutative ring. Suppose that every submodule of is generalized -radical supplemented, where is the injective hull of the simple module . Then is a uniserial ring.

Proof. Since is uniform, the hypothesis implies that is uniserial by Theorem 17. It follows from [13, Lemma 6.2] that is a uniserial ring.

Theorem 19. The following statements are equivalent for a ring .

(1) is semiperfect.(2)Every finitely generated free -module is generalized -radical supplemented.

Proof. By [14, Theorem 2.1] and Theorem 9, the proof is clear.

Lemma 20. Let be a module. Suppose that . Then is a generalized -radical supplemented module if and only if it is semisimple.

Proof. This is clear by [15, Proposition 3.3].

A ring is called left -ring if every simple left -module is injective. The ring is called an SSI ring if semisimple left -module is injective. Let be a commutative ring. is regular if and only if every simple left -module is injective [13, Theorem 2.14].

Proposition 21. Let be a left -ring and an -module. Then is generalized -radical supplemented if and only if is semisimple.

Corollary 22. Let be a commutative regular ring and an -module. Then is generalized -radical supplemented if and only if is semisimple.

Proposition 23. The following statements are equivalent for a ring .

(1)Every generalized -radical supplemented -module is injective. (2) is left Noetherian -ring.

Proof. By [16, Proposition 5.3], the proof is clear.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

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