Relative polynomial closure and monadically Krull monoids of integer-valued polynomials

TL;DR

This paper studies the structure of submonoids of integer-valued polynomial rings over Krull domains and valuation domains, introducing a modified polynomial closure concept to analyze their divisor properties.

Contribution

It constructs explicit divisor homomorphisms for submonoids of integer-valued polynomials and generalizes polynomial closure to subsets with finite polynomially dense subsets.

Findings

Submonoids are Krull monoids with explicit divisor homomorphisms.

Modified polynomial closure ensures finite polynomially dense subsets.

Results extend to integer-valued polynomials on subsets without isolated points.

Abstract

Let D be a Krull domain and Int(D) the ring of integer-valued polynomials on D. For any f in Int(D), we explicitly construct a divisor homomorphism from [f], the divisor-closed submonoid of Int(D) generated by f, to a finite sum of copies of (N_0,+). This implies that [f] is a Krull monoid. For V a discrete valuation domain, we give explicit divisor theories of various submonoids of Int(V). In the process, we modify the concept of polynomial closure in such a way that every subset of D has a finite polynomially dense subset. The results generalize to Int(S,V), the ring of integer-valued polynomials on a subset, provided S doesn't have isolated points in v-adic topology.