Commutative Monoid Rings as Hilbert Rings

Abstract

Let $S$ be a cancellative monoid with quotient group of torsion-free rank $\alpha$. We show that the monoid ring $R(S)$ is a Hilbert ring if and only if the polynomial ring $R[{\{ {X_i}\} _{i \in I}}]$ is a Hilbert ring, where $\left | I \right | = \alpha$.