For which rings $R$ is $\mathrm{SL}_n(R)$ generated by transvections?
I'm answering my own question based on the excellent reference given by Max and the additional comments of Jim Humphreys. There is nothing new in my answer, but I think it's useful to close the question in this way.
Following Hahn-O'Meara, we write $E_n(R)$ for the subgroup of $SL_n(R)$ generated by transvections (also called elementary matrices).
Theorem [H-O'M, Thm 4.3.9]. Let $R$ be a commutative ring. If $R$ is a Euclidean domain or a semilocal ring, then $SL_n(R) = E_n(R)$ for all $n$; If $R$ is a Hasse domain of a global field, then $SL_n(R) = E_n(R)$ for all $n \geq 3$ (and in many cases, but not always, also for $n=2$).
There are some other more general results known based on the so-called stable rank of the ring $R$, but as Jim pointed out, it seems hopeless to find a complete answer to the question.
The goal of my answer is only to provide recent references. I warmly recommend these two bits of T. Y Lam's book [2]:
- §I.8, for examples where transvections fail to generate $SL_n(R)$
- the second to last paragraph of §VIII.12 for other interesting examples of rings $R$ satisfying $SL_2(R) = E_2(R)$ or its negative.
And also B. Magurn's latest article on generalized Euclidean group rings [4].
Update.
Here are newer references focussing on the instances of $SL_2(R) \neq E_2(R)$ for $R$ a quadratic order in a totally imaginary quadratic field. The state of the arts is to be found in [3] and [6], while [5] gives a nice geometric insight on the set $SL_2(R)/E_2(R)$.
An older, but in my humble opinion, important paper is [1], where the structure of $SL_2(R)$ as an amalgamated product with factor $E_2(R)$ is described for $R$ the ring of integers of a totally imaginary quadratic field (with few exceptions), see Theorem 2.4.
[1] C. Frohman and B. Fine, "Some amalgam structure for Bianchi groups", 1988.
[2] T. Lam, "Serre's problem on projective modules", 2006.
[3] B. Nica, "The unreasonable slightness of $E_2(R)$ over imaginary quadratic rings", 2011.
[4] B. Magurn, "On a note from Oliver concerning generalized Euclidean group rings", 2014.
[5] K. Stange, "Visualizing the Arithmetic of Imaginary Quadratic Fields", 2017.
[6] A. Sheydvasser, "A Corrigendum to Unreasonable Slightness", 2017.
Further results are known: L. Vaserstein's paper "SL_2 of Dedekind rings of arithmetic type" proves these rings are generalized euclidean when they have a unit of infinite order. Integral group rings of finite groups are generalized euclidean when the group has no homomorphic image among the generalized quaternion groups of order a multiple of 4, no image among the binary polyhedral groups, and the abelianization of the group has generalized euclidean integral group ring. The finite abelian G with ZG euclidean include the cyclic groups, and Z/2 x Z/2, by the 1984 paper "Generalized euclidean group rings" by Dennis, Magurn & Vaserstein. But ZG is not generalized euclidean when SK_1(Z[G/[G,G]]) is non-vanishing, as it is for Z/4 x Z/2 x Z/2, for instance. So this is a delicate property!