Let $R$ be a commutative ring $R$ with $1$, and $n \geq 2$ an integer.



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> Under which conditions is the group $\operatorname{SL}\_n(R)$ generated by transvections?
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(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.



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).



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> 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 $n=2$).
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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.


