On generalized injective modules and almost injective modules

Abstract

A module [Formula: see text] is called [Formula: see text]-almost-invariant for a module [Formula: see text] if, for any homomorphism [Formula: see text], either [Formula: see text], or there exist nonzero direct summands [Formula: see text] of [Formula: see text] and [Formula: see text] of [Formula: see text] such that [Formula: see text] is an isomorphism and [Formula: see text], where [Formula: see text] and [Formula: see text] are the injective hulls of [Formula: see text] and [Formula: see text], respectively. This is a generalization of an almost [Formula: see text]-injective module. In this paper, we give a new characterization of a generalized [Formula: see text]-injective module by homomorphisms between their injective hulls, and consider conditions for an [Formula: see text]-almost-invariant module to be almost [Formula: see text]-injective. Moreover, we study a relationship between generalized [Formula: see text]-injective, almost [Formula: see text]-injective and [Formula: see text]-almost-invariant modules.