Duo Gamma Modules and Full Stability

In this work we study gamma modules which are implying full stability or implying by full stability. A gamma module is fully stable if for each gamma submodule of and each homomorphism of into . Many properties and characterizations of these classes of gamma modules are considered. We extend some results from the module to the gamma module theories.


1-Introduction:
In 1964, Nobusawa introduced the idea of gamma rings as a generalization of the idea of rings [1]. In 1966, Barnes summed up this idea and obtained entirety fundamental properties of gamma rings [2].
Let and be two additive abelian groups. is called a ring if there is a mapping ̅ ̅ such that the followings hold: (i) , (ii) (iii) and (iv) , for all . In 2010, Ameri and Sadeqhi extended the idea of modules to gamma modules [3]. Let be a ring. An additive abelian group is called a left odule, if there exists a mapping : , denote the image of such that the followings hold: , (iii) and (iv) , for all and .
In 1973, Faith introduced the definition of duo modules. Let be an module, a submodule of is said to be fully invariant if for each endomorphism of [4]. In the case that each submodule of is fully invariant, then is called duo.
In 1991, Abbas studied the relationship between the fully stable modules and the duo modules; an module is fully stable if for each submodule of , for each homomorphism from into [5].
In this paper, we consider the duo property in the category of gamma modules. A left module is called duo if for each submodule of and endomorphism of . For an arbitrary fixed in , a subset of and a subset of we define: and We give many properties and characterizations of this class of gamma modules. A left -module is a duo if and only if every cyclic submodule of is fully invariant where .We study the relationship between the duo and the multiplication gamma modules, while every fully stable gamma module is duo and the convers is true in principally quasi-injective gamma modules. We consider direct summand and sum of duo gamma modules. Finally, we consider some generalizations of full stability which are related to the duo property.

Basics of duo gamma modules
Let be an module. submodules of duo gamma modules may not be duo. However, every direct summand of duo gamma modules is a duo, for if is an submodule of a direct summand of an module and is an endomorphism of , then can be extended in the usual way to an endomorphism ̅ of , ̅ . It is clear that any fully stable module is a duo, but the converse is not true generally. For example, the module is a duo, but not fully stable. In the following, we consider conditions under which every gamma submodule of a duo module is a duo, as well as the homomorphic image, but first we introduce the following.
An module is said to be poorly injective, if each endomorphism of an submodule of can be extended to an endomorphism of . We call an module an quasi projective if, for any module and homo morphisms , : with is surjective, there is an endomorphism of such that . Then we have the following. . is a contradiction. Therefore is a duo.
It was previously proved [9] that a fully stable module satisfies for every pair of submodules of with . We have , but the converse may not be true. However, the converse is true in case that is fully essential stable [9]. In the following Lemma we have the following: Lemma (2.8): Let an module be a direct sum of submodules . Then is a fully invariant if and only if . Proof: Denote (resp. ) (resp. ) the canonical projection onto (resp. ) and : (resp. ) denote the injection mapping of (resp. Let be an module. An submodule of is called a essential if has a nontrivial intersection with every nonzero submodule of [10]. Dually, we say that an submodule of is called small if is a proper submodule of for each proper submodule of . An module is called Hopfian (resp. generalized Hopfian) if every surjective endomorphism of is an isomorphism (resp. has a small kernel). An module is called coHopfian (resp. weakly coHopfian) if every injective endomorphism of is an isomorphism (resp. has an essential image of ). Proposition (2.16): Every fully stable gamma module is a -coHopfian, and hence is a weakly -coHopfian. Proof: Let be a fully stable module and is an monomorphism, then . Hence, we have so that is an epimorphism. By Corollary (2.4) in a previous study [9], we have .

Proposition (2.17): Every duo gamma module is a generalized
Hopfian and a weakly coHopfian. Proof: Let be any surjective endomorphism of . Let such that . Then . It follows that is a small submodule of . Let be an injective endomorphism of , let such that . Since is fully invariant, we get and hence It follows that is an essential submodule of . Duo gamma modules are neither Hopfian nor coHopfian in general. We have seen in a previous article [9] that is a fully stable -module where is an arbitrary subring of . Let be an arbitrary fixed element in . The mapping , defined by for all in , is a surjective which is not an isomorphism, and hence is a duo which is not a Hopfian. On the other hand, it is clear that is a duo -module. We define by for all is an injective which is not an isomorphism.