Exercise 15, § 10

Let $A$ be a ring; for every integer $n \geqslant 1$, we identify canonically $A^n$ with the submodule of $A^{(\mathbf{N})}$ formed by the elements whose coordinates of index $\geqslant n$ are 0, and we denote by $A'_n$ the supplementary submodule formed by the elements whose coordinates of index $< n$ are 0. We identify every endomorphism $u \in \mathbf{GL}_n(A)$ with the automorphism of

A^{(N)} whose restriction to A^n is u and whose restriction to A'_n is the identity, so that $\mathbf{GL}_n(A)$ is identified with a subgroup of $\mathbf{GL}(A^{(N)})$; we denote by F the subgroup of $\mathbf{GL}(A^{(N)})$ union of the $\mathbf{GL}_n(A)$, in other words the subgroup leaving invariant all but a finite number of the elements of the canonical basis of $A^{(N)}$ (cf. II, p. 207, exerc. 12). Let T_n be the subgroup of $\mathbf{GL}_n(A)$ generated by the $B_{ij}(\lambda)$ for $0 \leq i < n,\ 0 \leq j < n,\ i \neq j,\ \lambda \in A$ (II, p. 161), and let T be the subgroup of F union of the T_n. (a) Now prove that for $n \geq 3$, T_n is equal to its commutator group. (b) Let a, b be two invertible elements of A. Now prove that one has:

where P, Q, P', Q' belong to T_2. Deduce that the matrix $\begin{pmatrix} aba^{-1}b^{-1} & 0 \\ 0 & 1 \end{pmatrix}$ belongs to T_2. (c) Deduce from b) that the commutator group of $\mathbf{GL}_n(A)$ is contained in T_{2n} (replace A by $\mathbf{M}_n(A)$ in b)). Conclude (using a)) that the commutator group of F is equal to T.