A Note on Dedekind and ZPI Modules

Let R be a domain. A non-zero R-module M is called a Dedekind module if every submodule N of M such that N M either is prime or has a prime factorization N=P1P2... PnN*, where P1, P2,... Pn are prime ideals of R and N* is a prime submodule in M. When R is a ring, a non-zero R-module M is called a ZPI module if every submodule N of M such that M either is prime or has a prime factorization. The purpose of this paper is to introduce interesting and useful properties of Dedekind and ZPI modules.