The Jacobson radical of commutative semigroup rings

Abstract

In this paper we consider semiprimitive commutative semigroup rings and related matters. A ring is said to be semiprhnitive if the Jacobson radical of it is equal to zero. This property is one of the most important in the theory of semigroup rings, and there is a prolific literature pertaining to the field (see 1,,14]). All semiprimitive rings are contained in another interesting class of rings. Let 8 denote the class of rings R such that ~/(R) = B(R), where J and B are the Jacobson and Baer radicals. Clearly, every semiprimitive ring is in 6. This class, appears, for example, in the theory of Pl-rings and in commutative algebra. (In particular, every finitely generated PI-ring and every Hilbert ring are in 6.) Therefore, it is of an independent interest. Meanwhile it is all the more interesting because any characterization of the semigroup rings in 6 will immediately give us a description of semiprimitive semigroup rings. Indeed, a ring R is semiprimitive if and only if R~6 and R is semiprime, i.e., B(R)=O. Semiprime commutative semigroup rings have been described by Parker and Giimer 1,12] and, in other terms, by Munn [9]. So it suffices to characterize semigroup rings in 6. Semigroup rings of 6 were considered by Karpilovsky r5], Munn 1,6-9], Okninski 1-10-h and others. In this paper commutative semigroup rings which are in 6 will be described completely. To this end one should know the structure of the Jacobson radical J(R[S]). In [2] Jespers described J(R1,S]) under rather weak assumptions on R. They hold, in particular, for every commutative R. Here we shall give another (quite short) description of J(R1,S]) which does not require any restriction on R. Besides, it is specially fitted for testing whether an element is in J(R1,S]), and this is essential for our proofs.