The subnormal structure of general linear groups

Abstract

Let A be an associative ring with 1. For any natural number n, let GLnA denote the group of invertible n by n matrices over A, and let EnA be the subgroup generated by all elementary matrices ai, j, where aεA and 1 ≤ i ≡ j ≤ n. For any (two-sided) ideal B of A, let GLnB be the kernel of the canonical homomorphism GLnA→GLn(A/B) and Gn(A, B) the inverse image of the centre of GLn(A/B) (when n > 1, the centre consists of scalar matrices over the centre of the ring A/B). Let EnB denote the subgroup of GLnB generated by its elementary matrices, and let En(A, B) be the normal subgroup of EnA generated by EnB (when n > 2, the group GLn(A, B) is generated by matrices of the form ai, jbi, j(−a)i, j with aA, b in B, i ≡ j, see [7]). In particuler,is the centre of GLnA