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.