Pi Semigroup Algebras of Linear Semigroups

Abstract

It is well-known that if a semigroup algebra K[S] over a field K satisfies a polynomial identity then the semigroup S has the permutation property. The converse is not true in general even when S is a group. In this paper we consider linear semigroups S C 9 (F) having the permutation property. We show then that K[S] has a polynomial identity of degree bounded by a fixed function of n and the number of irreducible components of the Zariski closure of S. A semigroup S is said to have the property 37, m > 2, if for every a,, .. ., am E S, there exists a non-trivial permutation a such that al a = a,()... aU(,nf). S has the permutation property 37 if S satisfies 3Y for some m> 2. The class of groups of this type was shown in [3] to consist exactly of the finite-by-abelian-by-finite groups. For the recent results and references on this extensively studied class of groups, we refer to [1]. The above description of groups satisfying 37 was extended to cancellative semigroups in [11], while a study of regular semigroups with this property was begun in [6]. In connection with the corresponding semigroup algebras K[S] over a field K, the problem of the relation between the property 37 for S and the Plproperty for K[S] attracted the attention of several authors. It is straightforward that S has 37 whenever K[S] satisfies a polynomial identity. However the converse fails even for groups in view of [3] and the characterization of PI group algebras, cf. [1 5]. On the other hand, K[S] was shown to be a PI-algebra whenever S is a finitely generated semigroup (satisfying 3 ) of one of the following types: periodic [20], cancellative [11], 0-simple [3, 5], inverse, or a Rees factor semigroup of free semigroup, cf. [12]. However, a finitely generated regular semigroup S with two non-zero OF-classes having Y but with K[S] not being PI was constructed in [12]. The main result of this paper is that if S is a linear semigroup satisfying 39, then K[S] is PI for any field K. In the course of the proof, we obtain a structural description of a strongly 7r-regular semigroup of this type. The basic technique is to consider the Zariski closure S of S. Then S is a linear Received by the editors July 7, 1989. 1980 Mathematics Subject Classification (1985 Revision). Primary 20M25, 16A38; Secondary 20M20, 16A45. ? 1990 American Mathematical Society 0002-9939/90 $1.00 + $.25 per page