Consider a commutative ring $x_ix_j = N_{ij}^k x_k$, where $N_{ij}^k \in\{0,1,2,3,\cdots\}$, where $\{x_i\}$ is a finite set. (This is actually a fusion ring and $x_i$ are simple objects.)

How to find the irreducible integer-matrix representations of the above ring? (ie what is the algorithm to find the irreducible representations given $N_{ij}^k$.)

(Given a group multiplication table of a finite group, there is a way to construct a (reducible) canonical integer-matrix representations by considering a vector space whose basis vectors are labelled by the group elements. Then such reducible representations contain all the irreducible representations. So we can try to decompose the canonical representations to find all the irreducible representations. Do we have similar results for integer-matrix representation of rings?)