Arithmetic functions on a Dedekind domain

TL;DR

This paper explores the structure of arithmetic functions on Dedekind domains, revealing their algebraic properties and how they uniquely characterize the domain through prime ideal analysis.

Contribution

It introduces a novel ring structure on functions from a monoid to a field and shows how totally multiplicative functions encode Dedekind domain isomorphism classes.

Findings

The ring of functions is isomorphic to a formal power series ring.

Totally multiplicative functions form a ringed space that characterizes Dedekind domains.

Prime ideal zeros determine the domain up to isomorphism.

Abstract

We study functions from a unique factorization monoid to a field. The set of all such functions is a commutative ring isomorphic to a ring of formal power series over the field, with indeterminates indexed by the prime elements of the monoid. The set of all totally multiplicative functions on the monoid of integral ideals in a Dedekind domain has a ringed space structure, which, after identifying functions with the same prime ideal zeros, determines the Dedekind domain up to isomorphism.