Semilocal group rings in characteristic zero

Abstract

It is shown that if F is a field of characteristic zero and G is a group such that the group ring F[G] is semilocal then G must be finite. A generalization to group rings over rings is given. A ring R is semilocal if R/J(R) is artinian, where J(R) denotes the Jacobson radical of R. R is said to be local if R/J(R) is a division ring. It is well known that the group ring is never local for a field of characteristic zero unless the group is trivial. As it is conjectured that J(F[G]) = (0) whenever F is a field of characteristic zero, we expect F[G] semilocal to imply G finite, and this is easily proved if F is not algebraic over the rationals, by a theorem of Amitsur [1]. The result in this paper may be interpreted as saying that the radical of the rational group ring cannot be too large. We would like to thank D. S. Passman for his helpful suggestions. LEMMA 1. Let K be a central subfield of a division ring D and let T E Mn (D), the full n by n matrix ring over D. Then the set S(T) = {k E K: 1k' T is singular} has at most n elements. PROOF. When written on the right, the elements of Mn(D) may be regarded as left D-linear transformations from the vector space Dn to Dn. If 1 k T is singular, there is a nonzero vector v E Dn such that v(l k'T) = 0, or vT = kv since k commutes with v. Hence k is an eigenvalue of T. Standard arguments of linear algebra show that eigenvectors in Dn corresponding to distinct eigenvalues of T in the centre of D are D-linearly independent. COROLLARY. Let R be a completely reducible K-algebra and let x E R. Then the set S(x) = {k E K: 1 k-x is not a unit in R} is finite. The following clever lemma forms a major part of the proof of the theorem in [3]. LEMMA 2 (FORMANEK). Let K be a subfield of the reals and let x = I_ aigi Received by the editors February 23, 1976. AMS (MOS) subject classifications (1970). Primary 16A26, 16A46; Secondary 16A10.