Group rings in which left ideal is a right ideal

Abstract

Let K[G] denote the group ring of G over the field K. In this note we characterize those group rings in which all left ideals are right ideals. Let R be a ring. We say that R is l.i.r.i. if every left ideal is a right ideal. A ring is l.a.r.i. if every left annihilator is a right ideal. Our notation follows that of [2]. The main results are THEOREM I. Let K be a field and let G be a nonabelian periodic group. Then if K[G] is l.a.r.i. one of the following occurs (i) Char K = 0 and G is a Hamiltonian group such that for each odd exponent, n, of G the quaternion algebra over the field K(4), where 4 is a primitive nth root of unity, is a division ring. (ii) Char K = 2 and K does not contain any primitive cube root of unity. Moreover G = Q x A, where Q is the quaternion group of order 8 and A is abelian in which each element has odd order and if n is an exponent for A, the least integer m > 1 satisfying 2m -1 (mod n) is odd. Conversely if K[G] satisfies either (i) or (ii), then K[G] is l.i.r.i. and, in particular, it is l.a.r.i. Observe that if char K > 2 and G is periodic, then K[GI is l.a.r.i. if and only if G is abelian. THEOREM II. Let K[G] denote the group ring over a nonabelian group. Then the following are equivalent (i) K[G] is l.i.r.i. (ii) G is locally finite and if a,f8 E K[GI with a,8 = 0, then /3a = 0. (iii) G is locally finite and K[G] is l.a.r.i. If we combine the above theorems we get necessary and sufficient conditions for K[GI to be l.i.r.i. By using the antiautomorphism of K[G] given by ,xeG k.x H* IeG kxxwe see that K[G] is l.i.r.i. (l.a.r.i.) if and only if K[G] is r.i.l.i. (r.a.l.i.). Received by the editors February 2, 1978. AMS (MOS) subject classifications (1970). Primary 16A26.