Primitivity of group rings of groups with non-trivial center (Algebras, logics, languages and related areas)

Abstract

In [2], we consider the following condition (*) : for each subset M of G consisting of a finite number of elements not equal to 1, and for any positive integer m, there exist distinct a, b, and c in G so that if (x_{1}^{-1}g_{1}x_{1})cdots(x_{m}^{-1}g_{rn}x_{m})=1, where g_{i} is in M and x_{i} is equal to a, b, or c for all i between 1 and m, then x_{i}=x_{i+1} for some i. This condition is often satisfied by a non-noetherian group which has a non-abelian free subgroup and the trivial center. For a such group G, we have proved that the group ring RG of G over a domain R is primitive if G satisfies (*) and |R|leq|G|. However as long as we deal with groups satisfying (*), since the center of them are always trivial, we can say nothing about primitivity of group rings of groups with nontrivial center. In this note, we consider a more general condition than the above one and give a primitivity result for group rings of groups with non-trivial center.