Semigroup Algebras and Noetherian Maximal Orders,

In this paper we describe when a monoid algebra K[S] is a noetherian PI domain which is a maximal order. Our work relies on the study of the height one primes of K[S] and of the minimal primes of the monoid S and leads to a characterization purely in terms of S. It turns out that the primes P intersecting S plays a crucial role, and therefore we reduce the problem to certain “local” monoids SP, that is, monoids with only one minimal prime. However, we illustrate by examples that such monoids and their algebras are much more complicated than the discrete valuation rings arising in the commutative case. Our work is based on the results of Brown concerning the group ring case and of Anderson and Chouinard on commutative monoid rings. We also rely on the description of noetherian monoid algebras obtained earlier by the authors.