On Quasi Maximal Ideals of Commutative Rings

Abstract

Let $$R$$ be a commutative ring with $$1\ne 0$$. A proper ideal $$I$$ of $$R$$ is said to be a quasi maximal ideal if for every $$a\in R-I$$, either $$I+Ra=R$$ or $$I+Ra$$ is a maximal ideal of $$R$$. This class of ideals lies between 2-absorbing ideals and maximal ideals which is different from prime ideals. In addition to give fundamental properties of quasi maximal ideals, we characterize principal ideal UN-rings with $$\sqrt{0}^2=(0)$$, direct product of two fields, and Noetherian zero dimensional modules in terms of quasi maximal ideals.