A note on maximal non-prime ideals

The rings considered in this article are commutative with identity 1≠0. By a proper ideal of a ring R, we mean an ideal I of R such that I≠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⊆A and I≠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.