Uniform almost relative injective modules

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 U be a uniform module and V be a finite direct sum of indecomposable modules. Recently, Singh (2016), has determined some conditions under which U is almost V injective, which generalize Baba's result. In the present paper some more results in this direction are proved. A module M is said to be completely almost self-injective, if for any two subfactors A, B of M, A is almost B-injective. A necessary and sufficient condition for a module M to be completely almost self-injective is given. Using this, it is proved that a Von Neumann ring R is completely almost right self-injective if and only if Rsoc(RR) is semi-simple and every minimal right ideal of R is injective.