On y-closed Dual Rickart Modules

In this paper, we develop the work of Ghawi on close dual Rickart modules and discuss y-closed dual Rickart modules with some properties. Then, we prove that, if are y-closed simple -modues and if -y-closed is a dual Rickart module, then either Hom ( ) =0 or . Also, we study the direct sum of y-closed dual Rickart modules.


INTRODUCTION
A module is called a dual Rickart module if for every , then for some Equivalently, a module is a dual Rickart module if and only if for every , then [1]. A module is called a closed dual Rickart module, if for any is a closed submodule in [1]. Recall that a submodule of an -module is called a y-closed submodule of if is nonsingular [2]. It is known that every y-closed submodule is closed.
In this paper, we give some results on the y-closed dual Rickart modules.
In section 2, we give the definition of the y-closed dual Rickart modules with some examples and basic properties. Moreover, we prove that for two -modules and and let be a submodule of if is -y-closed dual Rickart module, then is -y-closed dual Rickart module, see proposition (2.4) .
In section 3, we study the direct sum of y-closed dual Rickart modules. Furthermore, we prove that, let be two -modules, such that if is -y-closed dual Rickart module, then -y-closed dual Rickart module, for every submodule of , see proposition (3.1) .
Throughout this article, is a ring with identity and is a unital left -module. For a left module , will denote the endomorphism ring of . as -module is singular, therefore is not a y-closed submodule of Thus is not yclosed dual Rickart module. Proposition 2.4: Let and be two -modules and let be a submodule of . If is -y-closed dual Rickart module, then is -y-closed dual Rickart module. Proof. Let be an -homomorphism and let be the inclusion map. Consider the map . Since is -y-closed dual Rickart module, then is a y-closed submodule of and hence is nonsingular. But is a submodule of , therefore is nonsingular and hence is a y-closed submodule of . Thus is -y-closed dual Rickart module. Definition 2.5: Let be an -module, than is called a y-closed simple if and are the only yclosed submodules of . Proposition 2.6: Let be an -module and let be a y-closed simple -module. If is -y-closed dual Rickart module, then either (1) Hom( , )=0 or (2) Every nonzero -homomorphism from to is an epimorphism. Proof. Assume that Hom( , ) ≠0 and let be a non-zero -homomorphism. Since isy-closed dual Rickart, then is y-closed submodule of . But is y-closed simple, therefore and is an epimorphism. Recall that an -module is called a Co-Quasi-Dedekind -module if every nonzero endomorphism of is an epimorphism, see [3, p2] be two -modules with the property that the sum of any two y-closed submodule of is a y-closed submodule of . The following statements are equivalent (a) is a y-closed dual Rickart module, (b) ∑ is y-closed submodule of where is a finitely generated left ideal of .

Proof.
⇒ Let be a finitely generated left ideal of . Since is a yclosed dual Rickart module, then ( ) is a y-closed submodule of But ( ) Hence ∑ is a y-closed submodule of . ⇒ Clear.
Recall that an -module is called a faithful module if , where { } see [5, p206]. Before we give our next result, let us recall that an -module is called dualizable if ≠ 0, see [6, p10] . Proposition 3.4: Let be a y-closed simple, faithful -modue. If is y-closed dual Rickart module. Then is divisible. Proof. Suppose that is y-closed simple, faithful and y-closed dual Rickart module. Let R must be commutative.
It is clear that is an R-homomorphism. Since is a y-closed dual Rickart module, then is a y-closed submodule of . Since is a faithful module, then But is y-closed simple, therefore . Thus is divisible. Recall that an -module is called ⁄ cancellation module if it is faithful and for any ideal of such that implies , see [7] . Proposition 3.5: Let be a faithful, finitely generated and y-closed simple -module, where is not a field. Then is not y-closed dual Rickart module. Proof. Assume that is a y-closed dual Rickart module and let such that . Define It is clear that is an epimorphism, then is a yclosed sbmodule of . Since is a faithful module, then But is an y-closed simple module, therefore . Since M is finitely generated and faithful, then is ⁄ cancellation, by [7]. So, which is a contradiction. Thus is not y-closed dual Rickart module. Proposition 3.6: Let be an -module such that is -y-closed dual Rickart module. Then every cyclic submodule of is a y-closed submodule. Proof. Suppose that is an -module such that -y-closed dual Rickart module and let . Define Let be the inclusion map. Consider the map . It is clear that Since is -y-closed dual Rickart, then is a y-closed submodule of . Thus is a y-closed submodule of . Recall that an -module is called y-extending if for any submodule of there exists a direct summand of such that is essential in and is essential in see [8] . Proposition 3.7: Let be a y-extending -module. If is -y-closed dual Rickart module, for every index set , then is a semisimple module. Proof. Let