Exercise 17, § 2

Let A be a ring, $n$ an integer, and $X\in \mathbf{G}\mathbf{L}_n(A)$.

a) Suppose that the ring A is local. Prove that there exist matrices $P,Q$ in $\mathbf{G}\mathbf{L}_n(A)$ that are products of matrices of the form $B_{ij}(\lambda )$ (II, §10, No. 13, p. 361) and an invertible element $\delta$ of A, such that we have PXQ = diag(1$, . . . ,1, \delta$ ) (follow the proof of Proposition 14 of loc. cit.).

b) Suppose that the ring A is Euclidean (VII, §1, p. 49, Exercise 7). Prove the same result.

c) If A is a local ring or a Euclidean commutative ring, then the group $\mathbf{S}\mathbf{L}_n(A)$ is generated by the matrices $B_{ij}(\lambda )$.

d) Suppose that A is local, and denote its maximal ideal by $\mathfrak{m}$. Prove that every matrix in $\mathbf{M}_n(A)$ with invertible image in $\mathbf{M}_n(A/\mathfrak{m})$ is itself invertible (use Lemma 1 of II, §10, No. 13, p. 361).