Arithmetic functions on a Dedekind domain

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.