Turkish Studies (Elektronik), vol.13, no.29, pp.89-94, 2018 (Peer-Reviewed Journal) identifier


In this work, weakly essential supplemented modules are definedand some properties of these modules are investigated. All rings will beassociative with identity and all modules will be unital left modules inthis work. Let M be an R-module. If every essential submodule of M hasa weak supplement in M, then M is called a weakly essentialsupplemented (or briefly weakly e-supplemented) module. Let M be anR-module and X≤M. If X is a weak supplement of an essentialsubmodule of M, then X is called a weak essential supplement (or brieflyweak e-supplement) submodule in M. Let M be an R-module, U be anessential submodule of M and M 1 ≤M. If M 1 is weakly e-supplementedand U+M 1 has a weak supplement in M, then U has a weak supplementin M. Using this we prove that the finite sum of weakly essentialsupplemented modules is weakly essential supplemented. It is provedthat every factor module and every homomorphic image of a weaklyessential supplemented module are weakly essential supplemented. LetM be a weakly essential supplemented module. Then M/RadM have noproper essential submodules. Let M be a weakly essential supplementedmodule. Then every finitely M-generated module is weakly essentialsupplemented. Let R be any ring. Then R R is weakly essentialsupplemented if and only if every finitely generated R-module is weaklyessential supplemented.