Is it possible to have a consistent definition of $A\bmod_{left} B$ that respects matrix multiplication from left where $A$ and $B$ are two square matrices with non-negative entries?
Is there a generalized Chinese Remainder Theorem?
We can have $$\lambda B=0\bmod_{left} B$$ $$(\lambda C +\mu D)\bmod B=\lambda C\bmod_{left} B +\mu D\bmod_{left} B$$ at every $\lambda,\mu\in\Bbb Z$.
To define $A=LB + C$ we can have $L$ to be the largest integer entried matrix such that $ij$th entry of $C$ lies in $[0,B_{ij}]$.
Similarly we can define $\bmod_{right}$ and $\bmod$ by $A=BR+C$ and $A=LBR+C$ respectively.
Could we impose any additional structure on $K$ and $C$ to make things work?
If all involved matrices are circulant then this works with each of the three definitions agreeing with each other.
Let $\mathbf D \in \mathbb Z^{N \times N}$ be a matrix with yet-to-be-determined properties. For any two matrices $\mathbf A, \mathbf B \in \mathbb Z^{N \times N}$ define $\mathbf A \equiv \mathbf B \pmod{\mathbf D}$ if there exists a matrix $\mathbf M \in \mathbb Z^{N \times N}$ such that $\mathbf A - \mathbf B = \mathbf D \mathbf M$
This is clearly an equivalence relation. In fact, if $\mathbf D = \operatorname{diag}(n_1,n_2,\dots,n_N)$, then $\mathbb Z^{N \times N}/\mathbf D$ is isomorphic to $\left( \prod_{i=1}^N \mathbb Z/n_i \mathbb Z \right)^N$
The only problem seems to be the non-negative entries requirement.