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On Property (A) of rings and modules over an ideal | ||
| Journal of Algebra and Related Topics | ||
| دوره 8، شماره 2، اسفند 2020، صفحه 57-74 اصل مقاله (304.72 K) | ||
| نوع مقاله: Research Paper | ||
| شناسه دیجیتال (DOI): 10.22124/jart.2020.16259.1197 | ||
| نویسندگان | ||
| S. Bouchiba* ؛ Y. Arssi | ||
| Department of Mathematics, Faculty of Sciences, University Moulay Ismail, Meknes, Morocco | ||
| چکیده | ||
| 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. | ||
| کلیدواژه‌ها | ||
| Amalgamated duplication؛ A-ring؛ zero divisor | ||
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آمار تعداد مشاهده مقاله: 786 تعداد دریافت فایل اصل مقاله: 956 |
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