Groups Acting on Finitely Generated Commutative Rings

Abstract

Let G be a polycyclic group, b an ideal of the group ring Z G and A an abelian normal subgroup of G. Put R=Z G/b and let S denote the subring of R generated by the image of A. Then S is a finitely generated commutative ring and G acts on S by conjugation and normalizes the image of A. We wish to work by induction. It is not sufficient to know about the group rings Z(G/A)≅ Z G/(A−1)Z G of G/A and Z A of A, say by induction on the Hirsch number. We also need to allow for how G acts on Z A and more generally on S. In this chapter we do the basic groundwork for this.