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 $\begin{pmatrix}a &b\\ c &d \end{pmatrix}\mapsto \begin{pmatrix}d &ct^{-1}\\ bt &a \end{pmatrix}$



For which rings R is SL_n(R) generated by its n-1 fundamental copies of SL_2(R)?
$SL(n, R)$ is indeed generated by the fundamental copies of $SL(2, R)$ when $R$ is a Bezout domain (or even a Hermitian ring), this is called [Smith normal form](https://en.wikipedia.org/wiki/Smith_normal_form). In fact, $SL(n, R)$ is not only generated by such subgroups, but is a product of $(n-1)^2$ fundamental copies of $SL(2, R)$. See [$SL\_2$-factorizations of Chevalley groups](http://dx.doi.org/%2010.1007/s10958-013-1145-8) for the proof of a slightly weaker bound of $n^2-n$ factors in case $R$ is a Bezout domain.


On the other hand, for any ring $R$ of stable rank at most $2$, one can express $SL(n, R)$ as a product of $\frac{1}{2}(3n^2-5n)$ copies of $SL(2, R)$. Here is a [link](https://arxiv.org/abs/1511.02906).


Even more, $SL(n, R)$ being generated by $SL(2, R)$'s is equivalent to the map $SL(2, R)/E(2, R)\to SL(n, R)/E(n, R)$ being surjective. The question whether it is so for a particular ring $R$ is known as the "surjective stability for $K\_1$", and there are lots of papers on the subject. To put it short, in terms of dimension of $R$ the best one can get is "stable rank at most 2", but there are plenty of examples of geometric, arithmetic or analytic nature having stable rank greater than 2 (or not known).


As for the last question, concerning the Kac—Moody groups of type $\widetilde{A}\_1$, there is a [paper](http://www.sciencedirect.com/science/article/pii/0021869390902116) by S. Sidki, but it deals with groups over fields.


