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 we try of decompose the canonical representations into irreducible representations. Do we have similar results for ring?)