On proper strong Property (𝒜) for rings and modules

Abstract

The main purpose of this paper is to introduce and investigate a new class of rings lying properly between the class of [Formula: see text]-rings and the class of [Formula: see text]-rings. The new class of rings, termed the class of [Formula: see text]-rings, turns out to share common characteristics with both [Formula: see text]-rings and [Formula: see text]-rings. Numerous properties and characterizations of this class are given as well as the module-theoretic version of [Formula: see text]-rings is introduced and studied.