On transfer homomorphisms in commutative rings with zero-divisors

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.