PRUFER-LIKE DOMAINS AND THE NAGATA RING OF INTEGRAL DOMAINS

Abstract

A subring A of a Prufer domain B is a globalized pseudo-valuation domain (GPVD) if (i) A hooked right arrow 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 is an element of D[X], N = {f is an element of D[X] vertical bar c(f) = D}, and N-v = {f is an element of D[X] vertical bar c(f)(-1) = D }. In this article, we study when the Nagata ring D[X](N) (more generally, D[X](Nv)) 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](Nv) is a GPVD if and only if D is a t-GPVD and D[X](Nv) has Prufer integral closure, if and only if D[X] is a t-GPVD, if and only if each overring of D[X](Nv) 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 Prufer integral closure. We also give several examples of integral domains D such that D[X](Nv) is a GPVD.