Given a matrix $A\in GL\_n(\mathbb{Q})$. Can it be expressed as a product of two matrices $B,C$ with $B\in GL\_n(\mathbb{Z}[1/p])$ and $C\in GL\_{n}(\mathbb{Z}\_{(p)})$, where $ \mathbb{Z\_p}$ denotes the localization of $\mathbb{Z}$ at $(p)$ ?



decompositions of matrices over $\mathbb{Q}$
Yes, there exists such a decomposition. Multiplying $A$ by a suitable integer, we may assume that $A \in M\_n(\mathbf{Z})$ and $\det(A) \neq 0$. By the theory of elementary divisors, we have $A=\gamma\_1 D \gamma\_2$ with $\gamma\_1,\gamma\_2 \in SL\_n(\mathbf{Z})$ and $D$ is a diagonal matrix with nonzero integral entries. Now the result is true for $D$, and letting $D=M\_1 M\_2$ with $M\_1 \in GL\_n(\mathbf{Z}[1/p])$ and $M\_2 \in GL\_n(\mathbf{Z}\_{(p)})$, we see that $B=\gamma\_1 M\_1$ and $C=M\_2 \gamma\_2$ work.


