On rings whose atomic modules are injective

Abstract

We introduce a new class of rings close to von Neumann regular rings and V-rings. We call a ring $R$, a right atomic V-ring (or right AV-ring) if every atomic (in the sense of Zhou and Duns) right $R$-module is injective. We observe that a right semi-Artinian ring is a right AV-ring if and only if it is a right $\Sigma$-V-ring. Among other results, we show that every finite direct product of full matrix rings over Abelian regular rings is both a right and a left AV-ring. We also detect that there exist regular AV-rings (and hence $\Sigma$-V-rings) of infinite index (of nilpotency).