Semigroup algebras and Noetherian maximal orders (Q5936922)

Let \(K\) be a field and let \(S\) be a monoid. It is an unsolved problem to characterize when the semigroup algebra \(K[S]\) is a prime Noetherian maximal order. This paper makes important progress on this problem by describing when \(K[S]\) is a Noetherian PI domain that is a maximal order. If \(S\) is a submonoid of a torsionfree finitely generated Abelian-by-finite group, then \(K[S]\) is a Noetherian maximal order if and only if \(S\) satisfies the acc on left and right ideals, \(S\) is a maximal order in its group of quotients, and \(S_P\) has only one minimal prime ideal for each minimal prime \(P\) and \(S\). Interesting examples are given to illustrate that this situation is much more complicated than the commutative case.