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 $I\subseteq \Z(R)$ and where $I\nsubseteq \Z(R)$. Finally, we investigate the behavior of the Property ($\mathcal A$) along an ideal with respect to direct products.
Bouchiba, S., & Arssi, Y. (2020). On Property (A) of rings and modules over an ideal. Journal of Algebra and Related Topics, 8(2), 57-74. doi: 10.22124/jart.2020.16259.1197
MLA
S. Bouchiba; Y. Arssi. "On Property (A) of rings and modules over an ideal". Journal of Algebra and Related Topics, 8, 2, 2020, 57-74. doi: 10.22124/jart.2020.16259.1197
HARVARD
Bouchiba, S., Arssi, Y. (2020). 'On Property (A) of rings and modules over an ideal', Journal of Algebra and Related Topics, 8(2), pp. 57-74. doi: 10.22124/jart.2020.16259.1197
VANCOUVER
Bouchiba, S., Arssi, Y. On Property (A) of rings and modules over an ideal. Journal of Algebra and Related Topics, 2020; 8(2): 57-74. doi: 10.22124/jart.2020.16259.1197
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