6 - AN APPLICATION TO GROUP RINGS

Summary

This brief chapter is not really part of the theory of skew linear groups. The latter is too recent for any real applications to have arisen. However we show here how it sheds a little clarification on some known results concerning group rings.

Throughout this chapter F is a field of charactersitic p ≥ 0. A group G has finite endomorphiBm dimension over F if every irreducible FGmodule has finite dimension over its endomorphism ring. (Since any left G-module V becomes a right G-module via vg = g^–1v, v ∈ V, g ∈ G, there is no need to specify left or right.) Let Z_F denote the class of all such groups G. An alternative formulation is given by the following: a group G is in Z_F if and only if every primitive image of FG is Artinian. Z_F-groups seem first to have arisen in connection with injective modules, see Passman, Section 12.4, and they have been completely characterized grouptheoretically in the following cases: locally finite Z_F-groups (Hartley, see Hartley and Passman, Section 12.4), finitely genertaed solubleby-finite Z_F-groups (Snider), soluble-by-finite Z_F-groups for F not locally finite (also Snider).

A smaller class of groups than Z_F is the class Y_F of groups G with finite centred endomorphism dimension over F, meaning that every irreducible FG-module has finite dimension over the centre of its endomorphism ring.