Generalized ⊕-Radical Supplemented Modules

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.