On regular group rings

Abstract

Let G be a multiplicative group, K a commutative ring with unit, and K(G) the group ring of G with respect to K. We say that K(G) is regular if given an x in K(G), there is a y in K(G) such that xyx = x. Using a homological characterization of regular rings which was found independently by M. Harada [2, Theorem 5] and the author, we prove that if G is locally finite, then K(G) is regular if and only if K is regular and is uniquely divisible by the order of each element in G. More generally we show that if K(G) is regular, then G is a torsion group and K is a regular ring which is uniquely divisible by the order of each element in G. A nonhomological proof of these results has been given by J. McLaughlin (unpublished). In conclusion, we show that if G is a commutative group and K is a field of characteristic not dividing the order of any element in G, then the weak global dimension of K(G) equals the rank of G. For the most part we follow the conventions and notations in [']. Let R be a ring (with unit) and A a left R-module. The weak left dimension of A is defined as follows (see [1, Chapter VI, Exercise 3]):