ABSTRACT
This chapter discusses the basic classes of semigroups for which necessary and sufficient conditions for the PI-property of the corresponding semigroup algebras may be given. The permutational property alone does not seem to be the right way to attack the problem of characterizing the PI-property of semigroup algebras of finitely generated semisimple semigroups that are not groups. Since semisimple semigroups with PI-semigroup algebras must be completely semisimple, the chapter proposes a way of relating the PI-property of semigroup algebra to the PI-property of some group algebras. It also presents a particular case, in which the PI-property may be checked locally—through the principal factor algebras.
