Quasi-primry submodules satisfying the primeful property I

Abstract

Let $R$ be a commutative ring with identity and $M$ a unital $R$-module. In this article we extend the notion of quasi-primary ideals to submodules. A proper submodule $N$ of $M$ is called quasi-primary if whenever $rx\in N$ for $r\in R$ and $x\in M$, then $r\in \sqrt{(N:M)}$ or $x\in radN$ where $radN$ is the intersection of all prime submodules of $M$ containing $N$. Also, we say that a submodule $N$ of $M$ satisfies the primeful property if $M/N$ is a primeful $R$-module. For a quasi-primary submodule $N$ of $M$ satisfying the primeful property, $\sqrt{(N:M)}$ is a prime ideal of $R$. For the existence of a module-reduced quasi-primary decomposition, the radical of each term appeared in decomposition must be prime. We provide sufficient conditions, involving the saturation and torsion arguments, to ensure that this property holds as is valid in the ideal case. It is proved that for a submodule $N$ of $M$ over a Dedekind domain $R$ which satisffies the primeful property, $N$ is quasi-primary if and only if $radN is prime.