Abstract. We consider algebras over a field of characteristic zero, and prove that the Jacobson radical is homogeneous in every algebra graded by a linear cancellative semigroup. It follows that the semigroup algebra of every linear cancellative semigroup is semisimple.Recently it has been shown that several theorems concerning various ring constructions can be obtained in the more general situation of graded rings. One of the long-standing problems on group algebras is that of whether every group algebra over any field of characteristic zero is semiprimitive. The answer is known to be positive for linear groups. We obtain a graded analog of this result.Group algebras of linear groups have been explored by many authors. Passman and Zalesskii investigated the Jacobson radical and semiprimitivity of these algebras, see [16]. A systematic study of the semigroup algebras of linear semigroups has been started by Okniński and Putcha ([12], [13], [14]). In particular, in [14] the radicals of algebras of connected algebraic monoids were described. For algebras over a field of characteristic zero, it is shown in [13, §3] that the radical of every algebra of a linear semigroup is nilpotent, and the algebra of the full matrix semigroup is semiprimitive. For the full matrix semigroup M over a finite field F and for a field K of characteristic different from that of F , it follows from Fadeev's Theorem (see Kovács [10]) that the semigroup algebra KM is semiprimitive.Let S be a semigroup. An algebra R = s∈S R s is said to be S-graded if R s R t ⊆ R st for all s, t ∈ S. An ideal I of R is said to be homogeneous if I = s∈S I s , where I s = I ∩ R s .
Theorem 1. Let S be a cancellative linear semigroup. Then the Jacobson radical of every S-graded algebra over a field of characteristic zero is homogeneous.The radical of a (not necessarily cancellative) semigroup algebra over a field cannot contain nonzero homogeneous elements. Indeed, the factor of the semigroup
