ABSTRACT
This chapter discusses an important special class of polynomial identity-algebras. One may expect an intrinsic characterization of Azumaya semigroup algebras, and the characteristic of the base field only. This is an easy application of the basic results on tensor products, which are Azumaya algebras. The chapter describes Azumaya group algebras, and starts with the case in which the Azumaya group is infinite. A preparatory lemma allows a reduction to the case of finitely generated groups, where the local-global criterion for separability can be used. It is interesting to consider whether a separability idempotent can be given in the case of Azumaya group algebras in positive characteristics. The chapter shows that Azumaya semigroup algebras of cancellative semigroups are, in fact, central extensions of some Azumaya group algebras. Azumaya semigroup algebras of cancellative semigroups have a nice construction. The chapter closes with some observations on Azumaya semigroup algebras of arbitrary semigroups.
