WEAKLY ESSENTIAL G-SUPPLEMENTED MODULES

Celil NEBİYEV – Hasan Hüseyin ÖKTEN

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 M, then M 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.
Every weakly essential supplemented module is also 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 a weakly essential g-supplemented R-module. If every nonzero submodule of N is essential in N, then N is weakly g-supplemented. It is showed that every factor module and every homomorphic image of a weakly essential g-supplemented module are weakly essential g-supplemented. It is also proved that the finite sum of weakly essential g-supplemented modules is weakly essential g-supplemented. Let N be a weakly essential g-supplemented R-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. 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 an weak g-supplement submodule in N, then N is weakly essential g-supplemented.

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