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. and 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
Bouchiba, S. , and Arssi, Y. . "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
CHICAGO
S. Bouchiba and 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
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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