“The Group Ring of a Class of Infinite Nilpotent Groups” by S. A. Jennings

Abstract

This chapter is based on a seminal paper entitled “The Group Ring of a Class of Infinite Nilpotent Groups” by S. A. Jennings. In Sect. 6.1, we consider the group ring of a finitely generated torsion-free nilpotent group over a field of characteristic zero. We prove that its augmentation ideal is residually nilpotent. We introduce the dimension subgroups of a group in Sect. 6.2. These subgroups are defined in terms of the augmentation ideal of the corresponding group ring. We prove that the nth dimension subgroup coincides with the isolator of the nth lower central subgroup. This is a major result involving a succession of clever reductions where nilpotent groups play a prominent role. Section 6.3 deals with nilpotent Lie algebras. We show that there exists a nilpotent Lie algebra over a field of characteristic zero which is associated to a finitely generated torsion-free nilpotent group. As it turns out, the underlying vector space of this Lie algebra has dimension equal to the Hirsch length of the given group.