$\varphi$-prime submodules

Abstract

Let $R$ be a commutative ring with non-zero identity and $M$ be a unitary $R$-module. Let $\mathcal{S}(M)$ be the set of all submodules of $M$, and $\varphi :\mathcal{S}(M)\to \mathcal{S}(M)\cup \{\varnothing \}$ be a function. We say that a proper submodule $P$ of $M$ is a prime submodule relative to $\varphi$ or $\varphi$-prime submodule if $a\in R$, $x\in M$ with $ax\in P\setminus \varphi (P)$ implies that $a\in (P{:}_{R}M)$ or $x\in P$. So if we take $\varphi (N)=\varnothing$ for each $N\in \mathcal{S}(M)$, then a $\varphi$-prime submodule is exactly a prime submodule. Also if we consider $\varphi (N)=\{0\}$ for each submodule $N$ of $M$, then in this case a $\varphi$-prime submodule will be called a weak prime submodule. Some of the properties of this concept will be investigated. Some characterizations of $\varphi$-prime submodules will be given, and we show that under some assumptions prime submodules and ${\varphi }_1$-prime submodules coincide.

8-pages, To apper in Glasgow Mathematical Journal