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

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