The study of varieties of universal algebras2 which was initiated by Birkhoff in 1935, [2], has received considerable attention during the past decade; the question of particular interest being: “Which varieties have a finite basis for their laws?” In that paper Birkhoff showed that the laws of a finite algebra which involve a bounded number of variables are finitely based, so it is not altogether surprising that finite algebras have received their share of this attention.

1. In [2,5,6,7] a.o. several interpretations of the inequality for all such that were given and the following was proved.

For formations of finite soluble groups, the properties of Frattini closure and local defineability are known to be equivalent (see [2]). The investigations of Barnes and Gastineau-Hills [1] on the other hand reveal that although every Frattini closed formation of finite-dimensional soluble Lie algebras over an algebraically closed field of zero characteristic is local, without the algebraic closure condition the relationship between the two properties breaks down even for supersoluble Lie algebras. We are concerned here with the analogous problem for rings. The main results are;

Smith [6, Theorem 2.18] proved that if A is a ring which has a right artinian right quotient ring and G is a poly- (cyclic-or-finite) group, then the group ring AG has a right artinian right quotient ring. We give here a different proof (and a generalization) of this result using methods developed by Jategaonkar [3,4]. THEOREM. Let A be a ring which has a right artinian right quotient ring, and let G be a group which has a (transfinite) ascending normal series with each factor either finite or cyclic, but only a finite number of finite factors. Then AG has a right artinian right quotient ring.