Abstract
We show that in the group of infinite invertible column-finite matrices over an associative ring R, every element is a product of a row- and column-finite matrix and a unitriangular matrix. Moreover we prove that its subgroup of banded matrices is generated by strings (block-diagonal matrices, with finite blocks along the main diagonal).
Introduction
Let R be an associative ring with 1. Let GLc(∞, R) denote the group of infinite ℕ × ℕ column-finite matrices over R, and GLrc(∞, R) its subgroup of row- and column-finite matrices. A systematic study of normal subgroups of GLc(∞, R) in the case of division rings was initiated by A. Rosenberg. Research continued in works of Maxwell, Robertson, Arrell, Arrell and Robertson, Hausen, Thomas and others. We refer to for a comprehensive survey.
In this paper we are interested in results concerning generators of subgroups and a special form of an element in the case of an arbitrary associative ring of coefficients. We prove:
Theorem 1.1Every element of GLc(∞, R) is a product of an invertible row- and column-finite matrix and an upper unitriangular matrix.
The matrix a ∈ GLc (∞, R) is called n-banded if aij = 0 for all i, j such that |j – i| > n and either ai+n, i ≠ 0 or ai,i+n ≠ 0 for at least one index i.