Classification of Irreducible Z+-Modules of a Z+-Ring Using Matrix Equations

This paper aims to investigate and categorize all inequivalent and irreducible Z+-modules of a commutative unit Z+-ring A, equipped with set {1, x, y, xy} satisfying x2=1,y2=1 as a Z+-basis by using matrix equations, which was part of a call for a Special Issue about matrix inequalities and equations by Symmetry. If the rank of the Z+-module n≤2, we prove that there are finitely many inequivalent and irreducible Z+-modules, respectively, one and three. However, if n≥3, there is no irreducible Z+-module.