PI semigroup algebras of linear semigroups

Abstract

It is well-known that if a semigroup algebra K [ S ] K[S] over a field K K satisfies a polynomial identity then the semigroup S S has the permutation property. The converse is not true in general even when S S is a group. In this paper we consider linear semigroups S ⊆ M n ( F ) S \subseteq {\mathcal {M}_n}(F) having the permutation property. We show then that K [ S ] K[S] has a polynomial identity of degree bounded by a fixed function of n n and the number of irreducible components of the Zariski closure of S S .