ON ø-SEMIPRIME SUBMODULES
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Let $R$ be a commutative ring with non-zero identity and $M$ be a unitary $R$-module. Let $S(M)$ be the set of all submodules of $M$ and $\phi:S(M)\rightarrow S(M)\cup\{\emptyset\}$ be a function. We say that a proper submodule $P$ of $M$ is a $\phi$-semiprime submodule if $r\in R$ and $x\in M$ with $r^2x\in P\setminus \phi(P)$ implies that $rx\in P$. In this paper, we investigate some properties of this class of sub-modules. Also, some characterizations of $\phi$-semiprime submodules are given.
Abstract
Let $R$ be a commutative ring with non-zero identity and $M$ be a unitary $R$-module. Let $S(M)$ be the set of all submodules of $M$ and $\phi:S(M)\rightarrow S(M)\cup\{\emptyset\}$ be a function. We say that a proper submodule $P$ of $M$ is a $\phi$-semiprime submodule if $r\in R$ and $x\in M$ with $r^2x\in P\setminus \phi(P)$ implies that $rx\in P$. In this paper, we investigate some properties of this class of sub-modules. Also, some characterizations of $\phi$-semiprime submodules are given.
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