On transfer homomorphisms in commutative rings with zero-divisors

TL;DR

This paper investigates the transfer homomorphisms between monoids of regular elements in commutative rings with zero-divisors, especially Krull rings and their generalizations, to understand their arithmetic properties.

Contribution

It provides sufficient conditions for subrings of Krull rings to induce transfer homomorphisms, linking their arithmetic properties to those of the larger rings.

Findings

Established conditions for transfer homomorphisms between subrings and Krull rings.

Demonstrated that such transfer homomorphisms preserve many arithmetic properties.

Extended the study to generalizations like weakly Krull rings and C-rings.

Abstract

We study the arithmetic of monoids of regular elements of commutative rings with zero-divisors. Our focus is on Krull rings and on some of their generalizations (such as weakly Krull rings and C-rings). We establish sufficient conditions for a subring $R$ of a Krull ring $D$ guaranteeing that the inclusion $R^{\bullet} \hookrightarrow D^{\bullet}$ of the respective monoids of regular elements is a transfer homomorphism. The arithmetic of the Krull monoid $D^{\bullet}$ is well studied and the existence of a transfer homomorphism implies that $R^{\bullet}$ and $D^{\bullet}$ share many arithmetic properties.