On the maximal quotient ring of regular group rings (Q1101816)

Let K[G] be a von Neumann regular group ring of the group G over a field K. A special case of a result of Kaplansky [\textit{K. R. Goodearl}, Von Neumann regular rings (1979; Zbl 0411.16007)] states that any regular, right self-injective ring is uniquely a direct product of rings of types \(I_ f\), \(I_{\infty}\), \(II_ f\), \(II_{\infty}\), III. In this paper the author proves that the type \(I_ f\) part of the maximal right quotient ring Q r(K[G]) is non-zero if and only if \([G:\Delta (G)]<\infty\) and \(| \Delta (G)'| <\infty\). In this case, let M be the smallest normal subgroup of G with G/M abelian-by-finite. Then the type \(I_ f\) part of Q r(K[G]) is isomorphic to Q r(K[G/M]). \textit{J.-M. Goursaud} and \textit{J. Valette} [Bull. Soc. Math. Fr. 103, 91-102 (1975; Zbl 0309.16011)] proved some special cases of this result.