The canonical form of multiplication modules

Abstract

Let $R$ be a commutative ring with unit. An $R$-module $M$ is called a multiplication module if for every submodule $N$ of $M$, there is an ideal $I$ of $R$ such that $N=IM$. $M$ is called also a CF-module if there is some ideals $I_1,...,I_n$ of $R$ such that $M \simeq R/I_1 \bigoplus R/I_2 \bigoplus ... \bigoplus R/I_n$ and $I_1 \subseteq I_2 \subseteq ... \subseteq I_n$. In this paper, we use some new results about $\mu_R(M)$ the minimal number of generators of $M$ to show that a finitely generated multiplication module is a CF-module if and only if it is a cyclic module.