Krull Semigroups and Divisor Class Groups

Abstract

R. Matsuda has shown that a group ring is a Krull domain if and only if the coefficient ring is a Krull domain and the group is a torsion-free abelian group satisfying the ascending chain condition (ace) on cyclic subgroups [6]. D. F. Anderson has used this to obtain a partial determination of when a semigroup ring is a Krull domain, and under certain circumstances to describe the divisor class group of such a ring ([1], [2]). Using some of Anderson's techniques, but taking a different approach, we arrive at a complete answer of a different nature to these questions. We call a semigroup satisfying the major new conditions arising a Krull semigroup, and define its divisor class group.In particular, every abelian group is the divisor class group of such a ring, and it follows that every abelian group is the divisor class group of a quasi-local ring, which seems to be a new result.