**ESSENTIAL G-SUPPLEMENTED MODULES**

**STRUCTURED ABSTRACT**

In this work 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 M be an R-module. If every essential submodule of M has a g-supplement in M, then M is called an essential g-supplemented (or briefly ge-supplemented) module. Clearly we can see that every g-supplemented module is essential g-supplemented. Because of this essential g-supplemented modules are more generalized than g-supplemented modules. Every (generalized) hollow and every local module are essential g-supplemented. Let M be an essential g-supplemented R-module. If every nonzero submodule of M is essential in M, then M is g-supplemented. As every supplemented module is g-supplemented, every essential supplemented module is essential g-supplemented. It is proved that every factor module and every homomorphic image of an essential g-supplemented module are essential g-supplemented. Because of this essential g-supplemented modules is more generalized than essential supplemented modules. Let M be an R-module, U⊴M and N≤M. If N is essential g-supplemented and U+N has a g-supplement in M, then U has also a g-supplement in M. By using this we prove that the finite sum of essential g-supplemented modules is essential g-supplemented. Let M be an essential g-supplemented module. Then M/RadgM have no proper essential submodules. Let M be an essential g-supplemented R-module. Then every finitely M-generated R-module is essential g-supplemented. By using this we can prove the following Proposition: ‘Let R be any ring. Then RR is essential g-supplemented if and only if every finitely generated R-module is essential g-supplemented’. Let M be an R-module and V≤M. If V is a g-supplement of an essential submodule in M, then V is called an essential g-supplement (or briefly ge-supplement) submodule in M. Let M be an R-module. If every essential submodule of M is g* equivalent to an essential g-supplement (ge-supplement) submodule in M, then M is essential g-supplemented. By using this we can give following corollaries.

Corollary: Let M be an R-module. If every essential submodule of M is * equivalent to an essential g-supplement (ge-supplement) submodule in M, then M is essential g-supplemented.

Corollary: Let M be an R-module. If every essential submodule of M lies above an essential g-supplement (ge-supplement) submodule in M, then M is essential g-supplemented.

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

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

2010 Mathematics Subject Classification: 16D10, 16D70.