Given an $n \times n$ integral matrix $A \in M_n(\mathbb{Z})$, denote by $A_k \in M_n(\mathbb{Z}/p^k \mathbb{Z})$ its reduction modulo $p^k$. Now let $A, B \in M_n(\mathbb{Z})$ be be such that $A_k, B_k$ commute in $M_n(\mathbb{Z}/p^k\mathbb{Z})$. What can we say about $A$ and $B$?
I am looking for a statement of the form: there exist $A', B' \in M_n(\mathbb{Z})$ such that $A_k = A'_k, B_k = B'_k$ and $A'$ commutes with $B'$. Or maybe this is too much to ask but we can still find commuting $A', B'$ such that $A_l = A'_l, B_l = B'_l$, where $l$ is some function of $k$?
What if we further require that $A_k, B_k \in GL_n(\mathbb{Z}/p^k\mathbb{Z})$?
The question of whether almost-commuting matrices are close to commuting matrices has an enormous literature, but in everything I could find the matrices are complex and the "almost" and "close" is formalized in terms of the operator, Hilbert-Schmidt or Frobenius norm. Here I am asking the same question, but for integral matrices, where the "almost" and "closed" are formalized via equality modulo $p^k$.