On a generalization of prime submodules of a module over a commutative ring

Abstract

‎Let $R$ be a commutative ring with identity‎, ‎and $n\geq 1$ an integer‎. ‎A proper submodule $N$ of an $R$-module $M$ is called‎ ‎an $n$-prime submodule if whenever $a_1 \cdots a_{n+1}m\in N$ for some non-units $a_1‎, ‎\ldots‎ , ‎a_{n+1}\in R$ and $m\in M$‎, ‎then $m\in N$ or there are $n$ of the $a_i$'s whose product is in $(N:M)$‎. ‎In this paper‎, ‎we study $n$-prime submodules as a generalization of prime submodules‎. ‎Among other results‎, ‎it is shown that if $M$ is a finitely generated faithful multiplication module over a Dedekind domain $R$‎, ‎then every $n$-prime submodule of $M$ has the form $m_1\cdots m_t M$ for some maximal ideals $m_1,\ldots,m_t$ of $R$ with $1\leq t\leq n$‎.