TITLE: For which rings R is SL_n(R) generated by its n-1 fundamental copies of SL_2(R)?
QUESTION [15 upvotes]: By "fundamental copies" of $SL_2(R)$ in $SL_n(R)$, I mean those embedded along the diagonal (for instance, if $n=3$, those are the upper left and lower right corner copies of $SL_2(R)$ embedded in $SL_3(R)$).
When $R$ is a field or a Euclidean ring, $SL_n(R)$ is generated by elementary matrices. However, this need not be the case in general. My question is, whether one can nevertheless generate $SL_n(R)$ by its fundamental copies of $SL_2(R)$, even if $SL_n(R)$ is bigger than its elementary subgroup. I have the impression that the answer is at least "yes" when $R$ is a principal ideal domain or, more generally, a Bezout domain.
A similar question holds for the affine Kac-Moody group $SL_2(R[t,t^{-1}])$: when is it generated by its two fundamental copies of $SL_2(R)$, where the two embeddings of $SL_2(R)$ are given as follows:
$\begin{pmatrix}a &b\\ c &d \end{pmatrix}\mapsto \begin{pmatrix}a &b\\ c &d \end{pmatrix}$ and $\begin{pmatrix}a &b\\ c &d \end{pmatrix}\mapsto \begin{pmatrix}d &ct^{-1}\\ bt &a \end{pmatrix}$

REPLY [7 votes]: L. Vaserstein obtain a generation result in a very large generality, which encompasses all finitely generated commutative algebras over fields or PIDs:

Theorem (Vaserstein, 2006, unpublished): for every commutative noetherian ring that is finitely generated (by $k$ elements) over a subring of Krull dimension $\le 1$ and every $r\ge 3$, every matrix decomposes as a product of one matrix in the upper $\mathrm{SL}_2$ and a bounded (lazy bound: $\le 28k^2r^2$) number of elementary matrices.

(Note that obviously every elementary matrix is a product of a bounded (bound depending only on $r$) number of (elementary) matrices in one of the diagonal $\mathrm{SL}_2$.)
For somewhat absurd reasons, this paper is unpublished. I did in 2007 a detailed check of the proof, including a careful reading of his long 1976 paper with Suslin on which the paper strongly relies. Vaserstein's paper can be found on his web page: http://www.math.psu.edu/vstein/pm2.pdf; the above theorem follows from his Corollary 2.
Note that in case of finitely generated algebra over a field/PID, this is considerably more general than what is cited in Andrei's answer: the case of Bézout rings is strictly contained in the case of Krull dimension 1, and the case of stable $\le 2$ does not much go beyond Krull dimension 2: it covers the case of Krull dimension 1 and also maybe a few rings of Krull dimension 2. 
(Input welcome! I do not know if there's a commutative algebra that is finitely generated over a PID, has stable rank $\le 2$ and Krull dimension $\ge 2$. There is a MO post resulting in the conclusion that $\mathbf{Z}[x]$ has stable rank 3. According to this paper by Gabel (Pacific J. Math. 1975), polynomial rings over fields with $s\ge 4$ indeterminates have stable rank $\ge 3$, and for subfields of the real field this already holds when $s\ge 2$.)

Added: Vaserstein proved his result soon after Shalom proved and gave talks on a Kazhdan Property T result relying on the fact that for a ring $R$ of stable rank $d$, $\mathrm{SL}_r(R)$ for $r\ge d+1$ is boundedly generated by elementary matrices along with its upper left copy of $\mathrm{SL}_{d+1}(R)$. For $d=2$ this is of course the easy result on stable rank 2 retrieved later (link in Andrei's post). This is briefly mentioned in Shalom's ICM 2006 paper, with a reference to a paper which never appeared, and explained in detail in 2006/07 pdf notes of mine based on his talks. Combining Shalom and Vaserstein's result lead to the result that the subgroup generated by elementary matrices in $\mathrm{SL}_r(R)$ has Property T for every finitely generated commutative ring and $r\ge 3$. What happened a bit later is that the Property T result was obtained by other means, not using bounded generation, by Ershov and Jaikin-Zapirain (Inv. Math. 2010), in a wider generality ($R$ not necessarily commutative). This superseded the Property T result, but not Vaserstein's beautiful bounded generation result, which obviously is of interest independently of Kazhdan's Property T.