2-Prime ideals and their applications

This paper introduces the notion of 2-prime ideals, and uses it to present certain characterization of valuation rings. Precisely, we will prove that an integral domain R is a valuation ring if and only if every ideal of R is 2-prime. On the other hand, we will prove that the normalization (R) over bar of R is a valuation ring if and only if the intersection of integrally closed 2-prime ideals of R 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.