On the Nilpotency of the Jacobson Radical of Semigroup Rings

Munn [11] proved that the Jacobson radical of a commutative semigroup ring is nil provided that the radical of the coefficient ring is nil. This was generalized, for semigroup algebras satisfying polynomial identities, by Okniński [14] (cf. [15, Chapter 21]), and for semigroup rings of commutative semigroups with Noetherian rings of coefficients, by Jespers [4]. It would be interesting to obtain similar results concerning rings with nilpotent Jacobson radical. For band rings this was accomplished in [12], and for special band-graded rings in [13, §6]. However, for commutative semigroup rings analogous implication concerning the nilpotency of the radicals is not true: it follows from [7, Theorems 44.1 and 44.2], that if F is a field with charF = p and G is an infinite abelian p-group, then the Jacobson radical J(FG) is nil but not nilpotent. On the other hand, Braun [1] proved that the Jacobson radical of every finitely generated PI-algebra over a Noetherian ring is nilpotent. This famous result has several important corollaries (cf. [9], [19]). It shows that the existence of a finite generating set is a natural condition which may influence the nilpotency of the Jacobson radical of a ring. We shall prove the following