On Property (A) with respect to an ideal.

The main goal of this paper is to introduce and study the notion of Property (A) of a ring R or an R-module M with respect to an ideal I of R. The new notion turns out to be a weak form of the classic notion of an A-ring, in the sense that, any R-module satisfying the Property (A) satisfies as well the Property (A) with respect to any ideal I of R. Moreover, we prove that if I is contained in the nilradical of R, then the notion of A-module with respect to I and the notion of A-module collapse. Also, we present an example of a ring R possessing an ideal I ⊆ Z(R) such that R is an A-ring with respect to I while R is not an A-ring. Finally, we totally characterize the rings R and the R-modules M satisfying the Proprety (A) with respect to an ideal I as well as we investigate the behavior of the Property (A) with respect to an ideal vis-à-vis the direct products of rings and modules. [ABSTRACT FROM AUTHOR]