The radical of the algebra of any finite semigroup over any field

Abstract

For a finite semigroup S and a field Φ, denote by Φ [S] the semigroup algebra of S over Φ and when S has a zero element, denote by Φ0[S] the contracted semigroup algebra of S over Φ (see § 5.2 [1]). Then theorem 5.31 [1], due to E.Hewitt and H. S. Zuckerman, gives a determination of the radical of Φ0[S] when S is commutative and the characteristic of Φ does not divide the order of any subgroup of S; and on page 168 [1] some results concerning the radical of Φ0[S] when S is completely 0-simple are given, the determination of the radical in the general case remaining open.