ALMOST PRINCIPALLY SMALL INJECTIVE RINGS

Abstract

Let R be a ring and M a right R-module, S = <TEX>$End_R$</TEX>(M). The module M is called almost principally small injective (or APS-injective for short) if, for any a <TEX>${\in}$</TEX> J(R), there exists an S-submodule <TEX>$X_a$</TEX> of M such that <TEX>$l_Mr_R$</TEX>(a) = Ma <TEX>$Ma{\bigoplus}X_a$</TEX> as left S-modules. If <TEX>$R_R$</TEX> is a APS-injective module, then we call R a right APS-injective ring. We develop, in this paper, APS-injective rings as a generalization of PS-injective rings and AP-injective rings. Many examples of APS-injective rings are listed. We also extend some results on PS-injective rings and AP-injective rings to APS-injective rings.