University of GuilanJournal of Algebra and Related Topics2345-39318220201201On Property (A) of rings and modules over an ideal5774442110.22124/jart.2020.16259.1197ENS.BouchibaDepartment of Mathematics, Faculty of Sciences, University Moulay Ismail, Meknes, MoroccoY.ArssiDepartment of Mathematics, Faculty of Sciences, University Moulay Ismail, Meknes, MoroccoJournal Article20200414This 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 $I\subseteq \Z(R)$ and where $I\nsubseteq \Z(R)$. Finally, we investigate the behavior of the Property ($\mathcal A$) along an ideal with respect to direct products.https://jart.guilan.ac.ir/article_4421_33a8988d0fbb3d0d2247a464dd3299ee.pdf