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 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.