PrÜfer-Like Domains and the Nagata Ring of Integral Domains

Abstract

A subring A of a Prüfer domain B is a globalized pseudo-valuation domain (GPVD) if (i) A↪B is a unibranched extension and (ii) there exists a nonzero radical ideal I, common to A and B such that each prime ideal of A (resp., B) containing I is maximal in A (resp., B). Let D be an integral domain, X be an indeterminate over D, c(f) be the ideal of D generated by the coefficients of a polynomial f ∈ D[X], N = {f ∈ D[X] | c(f) = D}, and N v = {f ∈ D[X] | c(f)−1 = D}. In this article, we study when the Nagata ring D[X] N (more generally, D[X] N v ) is a GPVD. To do this, we first use the so-called t-operation to introduce the notion of t-globalized pseudo-valuation domains (t-GPVDs). We then prove that D[X] N v is a GPVD if and only if D is a t-GPVD and D[X] N v has Prüfer integral closure, if and only if D[X] is a t-GPVD, if and only if each overring of D[X] N v is a GPVD. As a corollary, we have that D[X] N is a GPVD if and only if D is a GPVD and D has Prüfer integral closure. We also give several examples of integral domains D such that D[X] N v is a GPVD.