In this paper, we study the class of modules having the property that if any pure submodule
is isomorphic to a direct summand of such a module then the pure submodule is itself a …

Recently, in a series of papers “simple” versions of direct-injective and direct-projective
modules have been investigated. These modules are termed as “simple-direct-injective” and …

A module M over a ring is called simple-direct-injective if, whenever A and B are simple
submodules of M with A≅ B and B⊆⊕ M, we have A⊆⊕ M. Various basic properties of …

In this paper, we introduce and study the dual notion of simple-direct-injective modules.
Namely, a right R-module M is called simple-direct-projective if, whenever A and B are …

The notion of simple-direct-injective modules which are a generalization of injective
modules unifies $ C2 $ and $ C3 $-modules. In the present paper, we introduce the notion of …

Abstract A right R-module M is virtually extending (or CIS) if every complement submodule of
M is isomorphic to a direct summand of M, and M is called a virtually C2-module if every …

Let R be a ring. We have studied rings whose projective modules have the extending
property of simple modules in [3] and [5]. In this note, we shall further study those rings when …

It is well known that over a left Notherian ring any direct sum of injecive modules is again
injective, and every injective module is a direct sum of indecomposable modules. Faith [6] …

In [9, Theorem 3.1] KR Fuller characterized indecomposable injective projective modules
over artinian rings using i-pairs. In [3] the author generalized this theorem to …

A ring $ R $ is regular [completely reducible] if and only if the character module of every left
$ R $-module is quasi-injective [quasiprojective]. Submodules of quasiprojective left $ R …