(untitled) (Journal of Mathematical Study, 2021)
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_+-basis1, 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.