strongly N-extending Strongly N-extending modules

Abstract

Relative extending modules and relative (quasi-)continuous modules were introduced and studied by Oshiro as a generalizations of extending modules and (quasi-) continuous respectively.  On other hand, Oshiro, Rizvi and Permouth introduced N-extending and N-(quasi-) continuous modules depending              where N and M are modules.  is closed under submodules, essential extension and isomorphic image. A module M is N-extending if for each submodule A , there is a direct summand B of M such that A is essential in B. Moreover, a module M is strongly extending if every submodule is essential in a stable (equivalently, fully invariant) direct summand of M. In this paper, we introduce and study classes of modules which are proper stronger than that of N-extending modules and N-(quasi-)continuous modules. Many characterizations and properties of these classes are given.