A well known consequence of the strong approximation theorem for semisimple simply connected algebraic groups over a number field is that certain reduction maps are surjective, for example, the canonical projection $Sp\_n({\mathbb Z}) \to Sp\_n({\mathbb Z}/m{\mathbb Z})$ is surjective for any modulus $m$. This latter fact has been proven by Newman and Smart (Acta Arithmetica 9, 1964) in a rather elementary way, and their proof appears to carry over to an arbitrary principal ideal domain (with an obvious modification in the proof of Lemma 1: Replace $E^+,E^-$ by the upper triangular part of the matrix $E$ (with zeros below the diagonal) and its transpose). 


I couldn't locate an explicit reference for this statement or similar results on surjectivity of reduction maps for other groups over a general principal ideal domain $R$ (or maybe other substitutes for the strong approximation theorem), except the review MR0865878 (88b:20072) of an article of Zhang and You in Dongbei Shida Xuebao (found via "Citations from reviews" in MathSciNet), which mentions in passing that the reduction map modulo an arbitrary $q$ for $Sp\_n(R)$ is onto.


The question then is: Does anybody know more references for this problem?



Strong approximation for principal ideal domains
For $G(R) = SL\_n(R)$ (see, e.g., this [MO post](https://mathoverflow.net/questions/78404/when-is-sln-r-rightarrow-sln-r-q-surjective/241596#241596)) or $G(R) = Sp\_{2n}(R)$, there is a common line of reasoning in order to prove that the reduction modulo $\mathfrak{a}$, say $\varphi\_{\mathfrak{a}}: G(R) \rightarrow G(R/\mathfrak{a})$, is surjective. It consists in showing that $G(R/\mathfrak{a})$ is generated by *elementary matrices* any of which has a lift in $G(R)$.


For $SL\_n(R)$, these elementary matrices are $E\_{ij}(r) = 1\_n + re\_{ij}$ with $1 \le i \neq j \le n$ and where $1\_n$ is the $n$-by-$n$ identity matrix and $e\_{ij}$ differs from the zero matrix only by its $(i, j)$ entry, which is $r \in R$.


For $Sp\_{2n}(R)$, these elementary matrices are 
$
SE\_{ij}(r) = \left\{
\begin{array}{ccc}
1\_{2n} + re\_{ij} & \text{ if } i = j',\\
1\_{2n} + re\_{ij} - (-1)^{i + j}re\_{j'i'} & \text{otherwise}.
\end{array}\right.
$
with $1 \le i \neq j \le n$ and
where $i \mapsto i'$ is the permutation of $\mathbb{N}$ defined by $(2i)' = 2i - 1$ and $(2i - 1)' = 2i$; alternatively, this is the infinite product of transpositions $(12)(34)(56) \cdots$.


If $SL\_n(R)$ is generated by the matrices $E\_{ij}(r)$ with $r \in R$, the ring $R$ is said to be $GE\_n$-ring in P. M. Cohn's terminology [1]. There are many examples of rings which are, or which are not $GE\_n$-rings, see [1, 3] or this [MO post](https://mathoverflow.net/questions/59884/for-which-rings-r-is-sl-nr-generated-by-transvections/240448#240448). I don't know about the terminology for $Sp\_{2n}(R)$, but here is a claim which addresses both groups.



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> > **Claim.** Let $R$ be a Noetherian one-dimensional domain, e.g., $R$ is a principal ideal domain. Let $G(R) = SL\_n(R)$ or $G(R) = Sp\_{2n}(R)$. Then $\varphi\_{\mathfrak{a}}$ is surjective for any ideal $\mathfrak{a}$ of $R$.
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> > *Proof.* We can assume that $\mathfrak{a} \neq 0$. Since $R/\mathfrak{a}$ is zero-dimensional, its Bass'stable rank is $1$. Therefore $G(R/\mathfrak{a})$ is generated by elementary matrices, see [1] and [2, Theorem 7.3.b]. Because any of these matrices has a lift in $G(R)$, the map $\varphi\_{\mathfrak{a}}$ is surjective.
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As explained by Marty, the above *should apply* more generally to simply connected Chevalley groups. Let $\Phi$ be a reduced irreducible root system, $R$ be a commutative ring, and let $G = G(\Phi, R)$ be the simply connected Chevalley group of type $\Phi$ over $R$. Let $E(\Phi, R)$ be the elementary subgroup of $G$.



> 
> 
> > 
> > **Claim to be checked**. Let $\Phi$ be a reduced irreducible root system and let $R$ be a Noetherian one-dimensional domain. Then $\varphi\_{\mathfrak{a}}$ is surjective for any ideal $\mathfrak{a}$ of $R$.
> > 
> > 
> > *Proof.* The condition on $R$ should be sufficient to ensure that 
> >  $G(\Phi, R/\mathfrak{a}) = E(\Phi, R/\mathfrak{a})$ for any non-zero ideal $\mathfrak{a}$.
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[1] P. Cohn, "On the structure of the $GL\_2$ of a ring", 1966.  

[2] L. Vaserstein, A. Suslin, "Serre's problem on projective modules over polynomial rings, and algebraic $K$-theory", 1976.  

[3] F. Grunewald, J. Mennicke and L. Vaserstein, "On the groups $SL\_2(\mathbb{Z}[x])$ and $SL\_2(k[x,y])$", 1994. 


