Let $R$ be a commutative ring $R$ with $1$, and $n \geq 2$ an integer.
Under which conditions is the group $\operatorname{SL}_n(R)$ generated by transvections?
(A transvection is a matrix with $1$ everywhere on the diagonal and exactly one other non-zero entry.) This is certainly the case if $R$ is a field, or if $R$ is a Euclidean domain, but I'm wondering whether there is a complete answer to the question.
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]:
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!
A nice account of the case
$$ n =2 $$
is given by I. Reiner in his review of a paper of P.M. Cohn below
The review is very detailed, Hope the tex may compile...
(that NOT worked !)
I give then just the review to try in MR:
MR0207856 (34 #7670) Cohn, P. M. On the structure of the ${\rm GL}_{2}$ of a ring. Inst. Hautes Études Sci. Publ. Math. No. 30 1966 5–53. 20.70 (16.48)
and the beginning of the review:
This well-written article encompasses a wealth of information about general linear groups over certain classes of rings. The author generalizes many earlier results about such groups, and gives a number of new and striking results. We proceed to describe some of the main theorems. Assume throughout that the underlying ring $R$ has a unity element and is associative, though not necessarily commutative. Denote by $U(R)$ its groups of units. (1) Let $\text{GL}_n(R)$ be the group of $n\times n$ invertible matrices over $R$, and $D_n(R)$ its subgroup of diagonal matrices. Let $E_n(R)$ be the group generated by the set of transvections $\{I+ae_{ij}\colon a\in R,1\leq i,j\leq n,i\neq j\}$, where $\{e_{ij}\}$ is a set of matrix units. Define $\text{GE}_n(R)=D_n(R)\cdot E_n(R)$, the subgroup of $\text{GL}_n(R)$ generated by elementary matrices. Of course, $E_n(R)\Delta\text{GE}_n(R)$. The author calls $R$ a generalized Euclidean ring (GE-ring) if $\text{GL}_n(R)=\text{GE}_n(R)$ for all $n$.
$$ \dots $$
