weakly supplement extending modules

Abstract

In this paper, the extending property of modules is generalized by using weakly supplement submodules. We call a module M is weakly supplement extending if each submodule of M is essential in a weakly supplement submodule of M. Many characterization of weakly supplement extending module are obtained, we show that M is weakly supplement extending if and only if each closed submodule is weakly supplementing submodule of M. Moreover, we study the relation of weakly supplement extending module and among other known classes of the module such as lifting module, weakly supplemented module, supplement extending module and others. Also, we study conditions under it a direct sum of weakly supplement extending module is weakly supplement extending.