Pseudo-valuation rings. (English) Zbl 0880.13011

Badawi, Ayman; Anderson, David F.; Dobbs, David E.

Cahen, Paul-Jean (ed.) et al., Commutative ring theory. Proceedings of the 2nd international conference, Fès, Morocco, June 5–10, 1995. New York, NY: Marcel Dekker. Lect. Notes Pure Appl. Math. 185, 57-67 (1997).

The aim of this paper is to generalize the theory of pseudo-valuation domains (PVD) to commutative rings with identity. The authors define a pseudo-valuation ring (PVR) as a ring where each prime ideal \(P\) is strongly prime, i.e. a prime ideal such that whenever \(a,b \in R\) then \(aP\) and \(bP\) are comparable. They extend known results on PVD, study the stability of the class of PVRs under passage to overrings and decide when the integral closure of a PVR \((R,M)\) equals \((M:M)\).
For the entire collection see [Zbl 0855.00015].

Reviewer: C.Galindo (Castellon)

MSC:

13F30	Valuation rings
13B22	Integral closure of commutative rings and ideals

Keywords:

pseudo-valuations domains; integral closure; pseudo-valuation ring