ON WEAKLY 2-ABSORBING PRIMARY SUBMODULES OF MODULES OVER COMMUTATIVE RINGS
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초록
Assume that $M$ is an $R$-module where $R$ is a commutative ring. A proper submodule $N$ of $M$ is called a weakly $2$-absorbing primary submodule of $M $ if $0\neq abm\in N$ for any $a,b\in R$ and $m\in M$, then $ab\in (N:M)$ or $am\in M\mbox{-rad}(N)$ or $bm\in M\mbox{-rad}(N).$ In this paper, we extended the concept of weakly $2$-absorbing primary ideals of commutative rings to weakly $2$-absorbing primary submodules of modules. Among many results, we show that if $N$ is a weakly $2$-absorbing primary submodule of $ M$ and it satisfies certain condition $0\neq I_{1}I_{2}K\subseteq N$ for some ideals $I_{1},I_{2}$ of $R$ and submodule $K$ of $M$, then $ I_{1}I_{2}\subseteq (N:M)$ or $I_{1}K\subseteq M\mbox{-rad}(N)$ or $ I_{2}K\subseteq M\mbox{-rad}(N)$.
Abstract
Assume that $M$ is an $R$-module where $R$ is a commutative ring. A proper submodule $N$ of $M$ is called a weakly $2$-absorbing primary submodule of $M $ if $0\neq abm\in N$ for any $a,b\in R$ and $m\in M$, then $ab\in (N:M)$ or $am\in M\mbox{-rad}(N)$ or $bm\in M\mbox{-rad}(N).$ In this paper, we extended the concept of weakly $2$-absorbing primary ideals of commutative rings to weakly $2$-absorbing primary submodules of modules. Among many results, we show that if $N$ is a weakly $2$-absorbing primary submodule of $ M$ and it satisfies certain condition $0\neq I_{1}I_{2}K\subseteq N$ for some ideals $I_{1},I_{2}$ of $R$ and submodule $K$ of $M$, then $ I_{1}I_{2}\subseteq (N:M)$ or $I_{1}K\subseteq M\mbox{-rad}(N)$ or $ I_{2}K\subseteq M\mbox{-rad}(N)$.
참고문헌 (15)
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