**WEAKLY ESSENTIAL G-SUPPLEMENTED MODULES**

**STRUCTURED ABSTRACT**

In this work weakly essential g-supplemented modules are defined and some properties of these modules are investigated. All modules are associative with unity and all modules are unital left modules. Let N be an R-module. If every essential submodule of N has a weak g-supplement in N, then N is called a weakly essential g-supplemented (or briefly weg-supplemented) module. Clearly we can see that every weakly g-supplemented module is weakly essential g-supplemented. Because of this weakly essential g-supplemented modules are more generalized than weakly g-supplemented modules. Weakly essential supplemented modules are weakly essential g-supplemented. Because of this weakly essential g-supplemented modules are more generalized than weakly essential supplemented modules. Every (generalized) hollow and every local module are weakly essential g-supplemented. Let N be an R-module and V≤N. If V is a weak g-supplement of an essential submodule in N, then V is called a weak essential g-supplement (or briefly weg-supplement) submodule in N. Let N be an R-module. If every essential submodule of N is a weak g-supplement in N, then N is weakly essential g-supplemented. By using this we clearly see that if every essential submodule of N is a weak essential g-supplement in N, then N is weakly essential g-supplemented. Let N be a weakly essential g-supplemented R-module. If every nonzero submodule of N is essential in N, then N is weakly g-supplemented. Moreover; N is also weakly supplemented. It is proved that every factor module and every homomorphic image of a weakly essential g-supplemented module are weakly essential g-supplemented. Let N be an R-module, U⊴N and K≤N. If K is weakly essential g-supplemented and U+K has a weak g-supplement in N, then U has also a weak g-supplement in N. By using this we show that the finite sum of weakly essential g-supplemented modules is weakly essential g-supplemented. Let N be a weakly essential g-supplemented module. Then N/RadgN have no proper essential submodules. Let N be a weakly essential g-supplemented R-module. Then every finitely N-generated R-module is weakly essential g-supplemented. By using this we can prove the following Proposition: ‘Let R be any ring. Then RR is weakly essential g-supplemented if and only if every finitely generated R-module is weakly essential g-supplemented’. Let N be an R-module. If every essential submodule of N is g* equivalent to a weak g-supplement submodule in N, then N is weakly essential g-supplemented. By using this we can give following corollaries:

Corollary: Let N be an R-module. If every essential submodule of N is * equivalent to a weak g-supplement submodule in N, then N is weakly essential g-supplemented.

Corollary: Let N be an R-module. If every essential submodule of N lies above a weak g-supplement submodule in N, then N is weakly essential g-supplemented.

Corollary: Let N be an R-module. If every essential submodule of N is g* equivalent to a g-supplement submodule in N, then N is weakly essential g-supplemented.

At the end of this work, we give an example separating with weakly essential g-supplemented and essential g-supplemented modules.

Keywords: Essential Submodules, Small Submodules, Supplemented Modules, G-Supplemented Modules.

2010 Mathematics Subject Classification: 16D10, 16D70.