Primitive group rings

Abstract

Two theorems showing the existence of primitive group rings are proved. THEOREM 1. Let G be a countable locally finite group and F a field of characteristic 0, or characteristic p if G has no elements of' order p. Then the group ring F[G] is primitive if and only if G has no finite normal subgroups. THEOREM 2. Let G be any grolp, and F afield. Then there is a g,roup H containing G suich that F[H] is a primitive ring. All rings will be associative and have a unit. R is a prime ring if xRy#O whenever x and y are nonzero elements of R. R is a (left) primitive ring if there is a faithful irreducible (left) R-module. Every primitive ring is prime, but not conversely. A group is locally finite if every finitely generated subgroup is finite. The prime group rings have been completely characterized by the following result: THEOREM 1 (CONNELL [1, p. 675]). The group ring R[G] is prime if and only if R is a prime ring and G has no finite normal subgroups. Very little is known about primitivity in group rings. Almost no progress has been made toward answering the general question, When is R[G] primitive? posed by Kaplansky [2] and Passman [3, p. 136]. Some negative statements are easy to make; for example, if R is a field and GO 1 is abelian or finite then R[G] is not primitive. Other negative results have been obtained by Alan Rosenberg [4]. But there were no examples of primitive R [G] with G #1. This paper proves two theorems in a positive direction. THEOREM 2. Suppose G is a countable locally finite group and F is a field of characteristic 0, or characteristic p if G has no elements of order p. Then F[G] is-primitive if and only if it is prime. THEOREM 3. Suppose G is a group and F is afield. Then there is a group H containing G such th'at F[H] is primitive. Received by the editors February 29, 1972 and, in revised form, April 28, 1972. AMS 1969 subject classifications. Primary 1642, 1653, 2080.