Weak Gamma (Quasi-) Continuous Modules

In this paper, we introduce and study the concept of weak gamma continuous modules as a generalization of continuous gamma modules.   A gamma submodule  of a gamma module   is called weak gamma summand if there exists submodule  of  and there exists an  ideal  of  where  and . An module   is called weak gamma extending module if every submodule  of  is essential in a weak gamma summand.  An module is called weak gamma continuous module if is it weak gamma extending module and if for each submodule  of  is isomorphic to a direct summand of , then  is weak gamma summand of .  Also it is called weak gamma quasi-continuous module if is weak gamma extending module and if  and  are direct summands of  with , then  is a weak gamma summand of . Many properties and results of the these modules are given


Introduction
An   −submodule  of   −module  is called a direct summand of , if there exists an   −submodule  of  such that  =  +  and  ∩  =  , write by  = ⨁ [6].An   −submodule  of  is called weak gamma summand (denoted by  ≤  ), if there exists   −submodule  of  and there exists ideal  of  such that  =  +  and  ∩  = .An   −module  is called  −extending module if every   −submodule of  is essential in a direct summand of  [3].As a generalization of  −extending module, an   −module  is called weak gamma extending module (shortly,  − module) if every   − submodule  of  is an essential in a weak gamma summand.An   −module  is called continuous gamma module if it is a  −extending module and if  ≅  ≤ ⨁ , then  ≤ ⨁  [6].We say that an   −module  is weak gamma continuous module if  is  −module and if for each   −submodule  of  is isomorphic to a weak gamma summand of , then  is a weak gamma summand of .An   −module  is called quasi-continuous gamma module if  is  −extending module and if  ≤ ⨁  and  ≤ ⨁  with  ∩  =  , then ⨁ ≤ ⨁  [6].As a generalization, an   − module  is called weak quasicontinuous gamma module if an   −module  is  −module and if  and  are direct summands of  with  ∩  =  , then ⨁ is weak gamma summand of  .An   − submodule  of an   − module  is called power essential gamma submodule ( −submodule) in  denoted by  ≤  , for each  ∈  and for each ideal  in  with  ≠ , then (:   ) ≠  [7].If  is ideal of  and  is   −submodule of , then  = {:  ∈ ,  ∈  and  ∈ } .We use  ≤  ,  ≤   ,  ≤ ⨁  ,  ≤   ,  ≤   ,  − module,  − module and  − module, for   − submodule, essential, direct summand, weak gamma summand, weak power essential gamma, gamma continuous module and gamma quasi-continuous module, respectively.For more basis refer to [1,2,4,5] and [7].

Weak Gamma Extending Modules
Definition (2.1):If  is an   −submodule of   −module , then we say that  is weak gamma summand of  (denoted by  ≤  ), if there exists an   −submodule  of  and an ideal  of  such that  +  =  and  ∩  = 0.In this case  =  ⊞ .
In the following proposition, we list some fundamental properties of weak gamma summand modules.

Definition(2.3):
We say that an   −module  is weak gamma extending module (for shortly  −module), if it satisfies ( 1 ): Every   −submodule  of   −module  is essential in a weak gamma summand.
It is clear that every gamma extending module is weak gamma extending module but the converse is not true, see

Examples (3.6)(1).
It is well known that every direct summand is closed, but weak gamma summand may not be closed in general see Examples (3.6) (7).
The following proposition show that if  is a weak gamma extending, then the closed   −submodule of  is weak gamma summand of .

Proposition(2.4):
If  is a weak gamma extending module, then every closed   −submodule of  is a weak gamma summand.
Proof: Suppose  is  −module, let  be closed   −submodule of , so  ≤   ≤   for some  ≤  but  is closed   −submodule of , then  = .

Proposition(2.5):
If every closed   −submodule of   −module  is weak gamma summand of , then  is weak gamma extending.
Proof: Let  be an   −submodule of , by Zorn's Lemma then  has a maximal essential extension such that  ≤   ≤  , so  is closed   − submodule of  , then  is weak gamma summand in  which implies that  ≤   ≤  .
Every direct summand of  −extending module is  −extending module [6].Examples (3.6)(8) shows that a weak gamma summand of weak gamma extending module may not be weak gamma extending module .

3-Weak Gamma (Quasi-)Continuous Module
We introduce the concept of weak gamma (quasi-)continuous module Definition(3.1):An   −module  is called weak gamma continuous module (shortly  −module) if it is has ( 1 ) and ( 2 ) where  2 means that if for each  ≅  ≤ ⨁  , then  ≤  .
Every continuous gamma module is weak continuous gamma module but converse not true, see Examples (3.6) (6).
It is clear that  −module is  −module, the converse not true see Examples (3.6) (6).
The prove of the following propositions direct [6].
2-Every simple   − module is  − module because every simple   − module is  − extending module, the converse is not true, for example  2 ⨁ 4 is  −module but it is not simple .