Almost relative injective modules

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