Exercise 15, § 10

Let $A$ be a ring; for every integer $n \geqslant 1$, $A^n$ is canonically identified with the submodule of $A^{(\mathbf{N})}$ consisting of the elements whose coordinates of index $\geqslant n$ are 0 and $A'_n$ denotes the complementary submodule consisting of the elements whose coordinates of index $< n$ are 0. Every endomorphism $u \in \mathbf{GL}_n(A)$ is identified with the automorphism of $A^{(\mathbf{N})}$ 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 $\mathbf{GL}(A^{(\mathbf{N})})$; let $F$ denote the subgroup of $\mathbf{GL}(A^{(\mathbf{N})})$ the union of the $\mathbf{GL}_n(A)$, in other words the subgroup leaving invariant the elements of the canonical basis of $A^{(\mathbf{N})}$ except for a finite number of them (cf. Exercise 12).

Let $T_n$ be the subgroup of $\mathbf{GL}_n(A)$ generated by the $B_{ij}(\lambda)$ for $0 \leq i < n, 0 \leq j < n, i \neq j, \lambda \in A$ (no. 13) and let $T$ be the subgroup of $F$ the union of the $T_n$.

(a) Show that for $n \geq 3, T_n$ is equal to its commutator group. (b) Let $a, b$ be two invertible elements of $A$. Show that

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 $T$.