Stable extending modules

Abstract EN

Let R be a ring and M be an R-module.

Recall that M is extends if, every sub module of M is essential in a direct summand of M.

Since 1980s, the developments of modules with the extending property have been a major area in ring and module theory.

In this paper, we introduce and study a proper generalization of extending modules.

We call an R-module M a stable extending if every stable sub module of M is essential in a direct summand of M.

Many characterizations and properties of stable extending modules are given.

Moreover, it is well known that a direct sum of extending modules need not be extending.

Unlike extending modules, we assert that a direct sum of stable extending modules is a stable extending.