Linear Groups over General Classes of Rings

Abstract

One of the important theorems of Chapter 2 asserts that if R is a division ring, then every proper normal subgroup of the elementary group E _n (R) is central (aside from two exceptions when n = 2), and consequently that the quotient of the elementary group E _n (R) by its center is a simple group. This result no longer holds for a general ring R, since any proper ideal of R gives rise to elementary congruence subgroups which are non-central, proper normal subgroups of E _n (R). Therefore the ideal structure of the ring R has direct impact on the normal subgroup structure of the group E _n (R) . In the case of the stable linear group we saw in Theorem 1.3.7 that the ideal structure of the ring classifies the normal subgroup structure of the group. In this chapter we will establish the analogue of this classification for the linear groups of finite rank. We will do so under the assumption that n is larger than both 2 and the stable rank of R. The stable rank of R is the smallest positive integer k such that for every m ≥ k, every unimodular vector of the module R ^m+1 can be reduced (in a way that will be made precise in §4.1A) to one in R ^m.