Let $\mathcal{O}$ be the ring of integers in an algebraic number field $K$ and let $M$ be a rank-$n$ projective $\mathcal{O}$-module. By definition, this means that $M \otimes K \cong K^n$, so the automorphism group of $M$ is a subgroup of $GL(n,K)$. It thus makes sense to talk about $SL(M)$.
Question: Is an explicit set of generators known for $SL(M)$?
If $M$ is free, then $SL(M) = SL(n,\mathcal{O})$ and much is known: Bass-Milnor-Serre proved that $SL(n,\mathcal{O})$ is generated by elementary matrices if $n \geq 3$, and Vaserstein proved that $SL(2,\mathcal{O})$ is generated by elementary matrices if $\mathcal{O}$ has infinitely many units. Of course, it is classical that $SL(n,\mathcal{O})$ is generated by elementary matrices if $\mathcal{O}$ is Euclidean.
