a note on maximal non-prime ideals

the rings considered in this article are commutative with identity $1neq 0$. by a proper ideal of a ring $r$,  we mean an ideal $i$ of $r$ such that $ineq r$.  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 $ isubseteq a$ and $ineq a$ 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 $r$.