Almost relative injective modules

Abstract

The concept of a module $M$ being almost $N$-injective, where $N$ is some module, was introduced by Baba (1989). For a given module $M$, the class of modules $N$, for which $M$ is almost $N$-injective, is not closed under direct sums. Baba gave a necessary and sufficient condition under which a uniform, finite length module $U$ is almost $V$-injective, where $V$ is a finite direct sum of uniform, finite length modules, in terms of extending properties of simple submodules of $V$. Let $M$ be a uniform module and $V$ be a finite direct sum of indecomposable modules. Some conditions under which $M$ is almost $V$-injective are determined, thereby Baba's result is generalized. A module $M$ that is almost $M$-injective is called an almost self-injective module. Commutative indecomposable rings and von Neumann regular rings that are almost self-injective are studied. It is proved that any minimal right ideal of a von Neumann regular, almost right self-injective ring, is injective. This result is used to give an example of a von Neumann regular ring that is not almost right self-injective.