On Property (A) of rings and modules over an ideal

This paper introduces and studies the notion of Property (A) of a ring R or an R-module M along an ideal I of R. For instance, any module M over R satisfying the Property (A) do satisfy the Property (A) along any ideal I of R. We are also interested in ideals I which are A-module along themselves. In particular, we prove that if I is contained in the nilradical of R, then any R-module is an A-module along I and, thus, I is an A-module along itself. Also, we present an example of a ring R possessing an ideal I which is an A-module along itself while I is not an A-module. Moreover, we totally characterize rings R satisfying the Property (A) along an ideal I in both cases where I⊆\Z(R) and where I⊈\Z(R). Finally, we investigate the behavior of the Property (A) along an ideal with respect to direct products.