A note on maximal non-prime ideals

The rings considered in this article are commutative with identity ۱\neq ۰. By a proper ideal of a ring R,  we mean an ideal I of R such that I\neq R.  We say that a proper ideal I of a ring R is a  maximal non-prime ideal if I is not a prime ideal of R but any proper ideal A of R with I\subseteq A and I\neq A is a prime ideal. That is, among all the proper ideals of R, I is maximal with respect to the property of being not a prime ideal.  The concept of maximal non-maximal ideal and maximal non-primary ideal of a ring can be similarly defined. The aim of this article is to characterize ideals I of a ring R such that I is a maximal non-prime (respectively, a maximal non maximal, a maximal non-primary) ideal of R.

Keywords:

Maximal non-prime ideal , maximal non-maximal ideal , maximal non-primary ideal , maximal non-irreducible ideal