When Is the Semigroup Ring Perfect?

A characterization of perfect semigroup rings A[G] is given by means of the properties of the ring A and the semigroup G. It was proved in [10] that for a ring with unity A and a group G the group ring A[G] is perfect if and only if A is perfect and G is finite. Some results on perfectness of semigroup rings were obtained by Domanov [3]. He reduced the problem of describing perfect semigroup rings A[G] to checking that certain semigroup algebras derived from A[G] satisfy polynomial identities. Further, a characterization of such /"/-algebras over a field of characteristic zero was found in [2]. However, the obtained results are difficult to formulate and refer to some exterior constructions obscuring an insight into the properties of the semigroup. The purpose of this paper is to completely characterize perfect semigroup rings by means of the properties of the semigroup and the coefficient ring. Our approach is quite different from that of [3] and omits /'/-methods. It works in arbitrary characteristic and the final result is a natural strengthening of the conditions for A[G] to be semilocal [7]. In what follows A will be an associative ring, G—a semigroup. A is said to be right perfect if it is semilocal with its Jacobson radical J( A ) (right) T-nilpotent (cf. [4]). By E(G) we shall mean the set of idempotents of G. If e G E(G), then we put Ge= [g E eGe \ g is invertible in eGe}. In the sequel the following well-known facts on T-nilpotence and perfectness will be useful. Lemma 1 (cf. [3,4]). I. If G is a nil semigroup with d.c.c. on right principal ideals, then G is T-nilpotent. 2. If H is an ideal in G and G has d.c.c. on right principal ideals, then the semigroups H, G/H also have d.c.c. on right principal ideals. 3. If H is an ideal in G, then A[G] is perfect if and only if so are the rings A[H],