The Chinese remainder theorem holds for Dedekind domains, and implies that if $\mathfrak{p}_1,...,\mathfrak{p}_n$ are prime ideals of a Dedekind domain $R$, and $\{a_1,...,a_n\}\subset frac(R)$, $\{r_1,...,r_n\}\subset\mathbb{Z}$, then there exists $b$ such that
- $b-a_i\in\mathfrak{p}_i^{r_i}R_{\mathfrak{p_i}}$ for $i=1,...,n$, and
- $b\in R_\mathfrak{p}$ for all $\mathfrak{p}\ne\mathfrak{p}_i$, for $i=1,...,n$.
I have seen this called 'strong approximation'. In this textbook, a strong approximation theorem is proved for quaternion algebras over global fields, and is said to hold for central simple algebras (except for totally definite quaternion algebras).
My question is: does strong approximation for central simple algebras imply a result in their maximal orders of the kind stated above for Dedekind domains? That is, given a sequence of elements and a collection of prime ideals in a maximal order, can one find an element congruent to the elements of the sequence modulo the prime ideals?