On the radical of group rings

Abstract

It is conjectured that, for any group G, the Jacobson radical \(J({\mathbb {Z}}[G])\) of the integral group ring \({\mathbb {Z}}[G]\) is zero. This is known to be true when G is finite. Here we show it is true for a reasonably large class of infinite groups, including finitely generated linear groups and groups which satisfy Higman’s ‘two unique products’ condition.