Non-Negative Integer Matrix Representations of a Z+-ring

The Z(+)-ring is an important invariant in the theory of tensor category. In this paper, by using matrix method, we describe all irreducible Z(+)-modules over a Z(+)-ring A, where A is a commutative ring with a Z(+)-basis {1, x, y, xy} and relations: x(2)=1, y(2)=1+x+xy. We prove that when the rank of Z(+)-module n >= 5, there does not exist irreducible Z(+)-modules and when the rank n <= 4, there exists finite inequivalent irreducible Z(+)-modules, the number of which is respectively 1, 3, 3, 2 when the rank runs from 1 to 4.