Noetherian semigroup algebras and prime maximal orders

Let S be a semigroup and K be a field. A K-space K[S], with basis S and with multiplication extending, in a natural way, the operation on S, is called a semigroup algebra. It remains an open problem to characterize semigroup algebras that are a prime Noetherian maximal order. In this thesis, we give an answer to the problem for a large class of cancellative semigroups and we illustrate these results with several examples of concrete classes of Noetherian maximal orders. Indeed, we find necessary and sufficient conditions for a prime Noetherian algebra K[S] of a submonoid S of a polycyclic-by-finite group to be a maximal order. Under an invariance condition on the minimal primes, our result is entirely in terms of the monoid S and, in order to prove it, we describe the height one prime ideals of K[S]. Recall that it is conjectured that polycyclic-byfinite groups G are the only groups having a Noetherian group algebra and K.A. Brown characterized when these group algebras K[G] are a prime Noetherian maximal order. In case K[S] also satisfies a polynomial identity, this means in case S is a submonoid of a finitely generated abelian-by-finite group, we show that the invariance condition on the minimal primes of S is necessary for K[S] to be a prime Noetherian maximal order. Furthermore, we establish a general method for constructing non-abelian maximal order semigroup algebras of finitely generated submonoids of abelian-by-finite groups, starting from abelian maximal orders. To obtain concrete constructions, we thus also need to deal with abelian finitely presented monoids A. If A has a presentation with one or two defining relations, we determine necessary and sufficient conditions for K[A] to be a domain that is a maximal order. The description is only in terms of the defining relations. Furthermore, we compute the class groups of such semigroup algebras. In the appendix, we briefly explain applications of maximal orders in space-time coding. These applications softly point out that maximal orders not only might be interesting for experts in algebra, but also for specialists in coding theory.