Weakly primal submodules

Abstract

Let $R$ be a commutative ring and let $M$ be an $R$-module. A submodule $N$ of $M$ is called a weakly primal submodule provided that the set $ P = w(N) \cup \{ 0 \} $ forms an ideal of $R$. Here $w(N)$ is the set of elements of $R$ that are not weakly prime to $N$, where an element $ r \in R $ is not weakly prime to $N$ if $ 0 \neq rm \in N $ for some $ m \in M \backslash N $. In this paper we give some basic results about weakly primal submodules. Also we discuss on the relations between the classes of the weakly primal submodules of $M$ and the weakly primal submodules of modules of fractions of $M$.