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 toA 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 alsogive several examples of integral domains D such that D X N v is a GPVD.

Topological properties of semigroup primes of a commutative ring

“…It is clear that the spectral map g a : Spec(R(T )) → Spec(R) is surjective. For uses of Nagata rings and related rings of rational functions in the context of star and semistar operations, see [28], [24], [25], [4], [5], [6], [7], [12], [34], [38], [37] and [45]. Now, we consider another map γ : Spec(R) → Spec(R(T )) by setting γ(P ) := P R(T ) for each P ∈ Spec(R): this map is well-defined and injective (since IR(T ) ∩ R = I, for all ideals I of R [28, Proposition 33.1(4)]).…”