Presentations and module bases of integer-valued polynomial rings

TL;DR

This paper investigates the structure of integer-valued polynomial rings over various domains, providing algebraic presentations and characterizations, especially for Krull and UFD domains, using the t-closure operation and module bases.

Contribution

It introduces a D-algebra presentation of Int(D^X) via generators and relations for a broad class of domains, including UFDs and Krull domains with regular bases, and characterizes when these rings are quotients of such polynomial rings.

Findings

Established D-algebra presentations for Int(D^X) for many domains.

Provided an intrinsic characterization of D-algebras isomorphic to quotients of Int(D^X).

Generalized the criterion for Krull domains to have a regular basis based on the Polya-Ostrowski group.

Abstract

Let D be an integral domain with quotient field K. For any set X, the ring Int(D^X) of integer-valued polynomials on D^X is the set of all polynomials f in K[X] such that f(D^X) is a subset of D. Using the t-closure operation on fractional ideals, we find for any set X a D-algebra presentation of Int(D^X)$ by generators and relations for a large class of domains D, including any unique factorization domain D, and more generally any Krull domain D such that Int(D) has a regular basis, that is, a D-module basis consisting of exactly one polynomial of each degree. As a corollary we find for all such domains D an intrinsic characterization of the D-algebras that are isomorphic to a quotient of Int(D^X) for some set X. We also generalize the well-known result that a Krull domain D has a regular basis if and only if the Polya-Ostrowski group of D (that is, the subgroup of the class group of D generated by the images of the factorial ideals of D) is trivial, if and only if the product of the height one prime ideals of finite norm q is principal for every q.