2-Prime ideals and their applications

Abstract

This paper introduces the notion of [Formula: see text]-prime ideals, and uses it to present certain characterization of valuation rings. Precisely, we will prove that an integral domain [Formula: see text] is a valuation ring if and only if every ideal of [Formula: see text] is [Formula: see text]-prime. On the other hand, we will prove that the normalization [Formula: see text] of [Formula: see text] is a valuation ring if and only if the intersection of integrally closed 2-prime ideals of [Formula: see text] is a 2-prime ideal. At the end of this paper, we will give a generalization of some results of Gilmer and Heinzer by studying the properties of domains in which every primary ideal is an integrally closed 2-prime ideal.