The Coefficient Ring of a Primitive Group Ring

Abstract

All rings are associative with unity. A ring R is prime if xRy ≠ 0 whenever x and y are nonzero. A ring R is (left) primitive if there exists a faithful irreducible left R-module.If the group ring R[G] is primitive, what can we say about R? First, since every primitive ring is prime, we know that R is prime, by the followingTHEOREM 1 (Connell [1, 675]). The group ring R[G] is prime if and only if R is prime and G has no non-trivial finite normal subgroup.