Some results on graded $S$-strongly prime submodules
محورهای موضوعی : Commutative algebra
1 - Department of Mathematics, Payame Noor University, P.O.BOX 19395-3697 Tehran, Iran
کلید واژه: Graded S-prime submodule, graded S-strongly prime submodule, graded multiplication module,
چکیده مقاله :
Let $G$ be a group with identity $e$ and $R$ be a commutative $G$-graded ring with nonzero identity, $S\subseteq h(R)$ a multiplicatively closed subset of $R$ and $M$ be a graded $R$-module. A graded submodule $N$ of $M$ with $(N:_{R}M)\cap S=\emptyset$ is said to be graded $S$-strongly prime if there exists $s\in S$ such that whenever $((N+Rx_{g}):_{R}M)y_{h}\subseteq N$, then $sx_{g}\in N$ or $sy_{h}\in N$ for all $x_{g},y_{h}\in h(M)$. The aim of this paper is to introduce and investigate some basic properties of the notion of graded $S$-strongly prime submodules, especially in graded multiplication modules. Moreover, we investigate the behaviour of this structure under graded module homomorphisms, localizations of graded modules, quotient graded modules, Cartesian product.
Let $G$ be a group with identity $e$ and $R$ be a commutative $G$-graded ring with nonzero identity, $S\subseteq h(R)$ a multiplicatively closed subset of $R$ and $M$ be a graded $R$-module. A graded submodule $N$ of $M$ with $(N:_{R}M)\cap S=\emptyset$ is said to be graded $S$-strongly prime if there exists $s\in S$ such that whenever $((N+Rx_{g}):_{R}M)y_{h}\subseteq N$, then $sx_{g}\in N$ or $sy_{h}\in N$ for all $x_{g},y_{h}\in h(M)$. The aim of this paper is to introduce and investigate some basic properties of the notion of graded $S$-strongly prime submodules, especially in graded multiplication modules. Moreover, we investigate the behaviour of this structure under graded module homomorphisms, localizations of graded modules, quotient graded modules, Cartesian product.
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