Remarks on pseudo-valuation rings

A prime ideal P of a ring A is said to be a strongly prime ideal if aP and bA are comparable for all a,b epsilon A. We shall say that a ring A is a pseudo-valuation ring (PVR) if each nr;me ideal of A is a strongly prime ideal. We show that If A Is a PVR with maximal ideal M, then every overring of A is a PVR if and only if M is a maximal ideal of every overring of A that does not contain the reciprocal of any element of M. We show that if R is an atomic domain and a PVD, then dim(R) less than or equal to 1. We show that if R is a PVD and a prime ideal of R is finitely generated, then every overring of R is a PVD. We give a characterization of an atomic PVD in terms of the concept of half-factorial domain.