Abstract
We investigate the class groups and the distribution of prime divisors in affine monoid algebras over fields and thereby extend the result of Kainrath that every finitely generated integral algebra of Krull dimension at least 2 over an infinite field has infinitely many prime divisors in all classes.
Highlights
The study of class groups and of the distribution of prime divisors in the classes is an old topic in ring theory
For every abelian group G and every subset G0 ⊆ G, which generates G as a monoid, there is a Dedekind domain R whose class group is isomorphic to G and G0 corresponds to the set of classes containing prime divisors
It is classical that rings of integers in algebraic number fields and holomorphy rings in algebraic function fields are Dedekind domains with finite class group and infinitely many prime divisors in all classes
Summary
The study of class groups and of the distribution of prime divisors in the classes is an old topic in ring theory. Monoid algebras that are Krull and cluster algebras that are Krull do have infinitely many prime divisors in all classes ([6] and [8, Theorem A]). Concerning Noetherian domains, that are not necessarily Krull (equivalently, not integrally closed) we mention a result by Kainrath [13] He proved that for an infinite field K, every finitely generated integral K-algebra with quotient field seperable over K and Krull dimension at least 2 has infinitely many prime divisors in each class. In particular, this holds true for finitely generated monoid algebras of this type.
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