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S-small and S-essential submodules | ||
| Journal of Algebra and Related Topics | ||
| دوره 10، شماره 1، شهریور 2022، صفحه 1-10 اصل مقاله (295.66 K) | ||
| نوع مقاله: Research Paper | ||
| شناسه دیجیتال (DOI): 10.22124/jart.2021.20856.1328 | ||
| نویسنده | ||
| S. Rajaee* | ||
| Department of Mathematics, University of Payame Noor,Tehran, Iran. | ||
| چکیده | ||
| This paper is concerned with S-comultiplication modules which are a generalization of comultiplication modules. In section 2, we introduce the S-small and S-essential submodules of a unitary $R$-module $M$ over a commutative ring $R$ with $1\neq 0$ such that S is a multiplicatively closed subset of $R$. We prove that if $M$ is a faithful S-strong comultiplication $R$-module and $N\ll ^{S}M$, then there exist an ideal $I\leq ^{S}_{e}R$ and an $t\in S$ such that $t(0 :_{M}I)\leq N\leq (0 :_{M}I)$. The converse is true if $S\subseteq {\rm U}(R)$ such that ${\rm U}(R)$ is the set of all units of $R$. Also, we prove that if $M$ is a torsion-free S-strong comultiplication module, then $N\leq ^{S}_{e} M$ if and only if there exist an ideal $I\ll ^{S}R$ and an $s\in S$ such that $s(0 :_{M} I)\leq N\leq (0 :_{M} I)$. In section 3, we introduce the concept of S-quasi-copure submodule $N$ of an $R$-module $M$ and investigate some results related to this class of submodules. | ||
| کلیدواژه‌ها | ||
| S-small submodule؛ S-essential submodule؛ S-co-m module | ||
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