A note on maximal non-maximal prime submodule

Abstract

The rings considered in this article are commutative with identity 1 ≠  0 and the modules considered in this article are modules over commutative rings and are unitary. We say that a proper submodule N of an R-module M is a maximal non-prime submodule if N is not a prime submodule of M but any proper submodule K of M with N Í K and N ≠ K is a prime submodule. That is, among all the proper submodules of M, N is maximal with respect to the property of being not a prime submodule. The concept of maximal non-maximal submodule, maximal non-primary submodule and maximal non-irreducible submodule of a module can be similarly defined. The aim of this article is to characterize submodules N of an R-module M such that N is a maximal non-prime (respectively, a maximal non-maximal, a maximal non-primary, a maximal non-irreducible) submodule of M.