Almost relative injective modules

The concept of a moduleM being almostN-injective, whereN is some module, was introduced by Baba (1989). For a given module M , the class of modules N, for which M is almostN-injective, is not closed under direct sums. Baba gave a nece ssary and sufficient condition under which a uniform, finite le ngth moduleU is almost V-injective, whereV is a finite direct sum of uniform, finite length modules, in ter ms 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 condit i s under whichM is almostV-injective are determined, thereby Baba’s result is genera liz d. A module M that is almostM-injective is called an almost self-injective module. Comm utative 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 i s used to give an example of a von Neumann regular ring that is not almost right self-injec tive.