On the maximal quotient ring of regular group rings

Abstract

All rings considered in this paper are associative with 1, and all modules are unital. A ring R is (von Neumann) regular provided that for every x E R there exists y E R such that X~.Y = x. Recall that a special case of a result of Kaplansky [2, Theorem 10.221 states that any regular, right self-injective ring is uniquely a direct product of rings of Types I,, I,, III, II x , III. We refer the reader to [2] for undefined terms on regular rmgs. If R is a regular ring, then it is well known that its maximal right quotient ring Q’(R) is regular and right self-injective (see [2, Corollaries 1.2 and 1.241). If G is a group and K is a (commutative) field, we denote by K[G] the group ring of G over K. We refer the reader to [9] for the general theory on group rings. Let K[G] be a regular group ring. By following the general theme of Hartley’s paper [7], we are able to obtain the main result of this paper (Theorem 2.3), which states that the Type I/part of Q’(K[G]) is non-zero if and only if [G: d(G)] < 00 and Id(G)‘1 <co. In this case, let M be the smallest normal subgroup of G with G/M abelian-by-finite. Then the Type I, part of Q’(K[G]) is isomorphic to Qr( K[G/M]). Goursaud and Valette [3] proved some special cases of this result, namely when K either has positive characteristic or contains all roots of unity. It is well known that the maximal quotient ring of a regular ring whose irreducible modules are finite dimensional over their commuting rings is