The non-integer matrix (NIM) solutions of matrix equations are closely related to the NIM representations over fusion rings and the module categories over tensor categories. In this talk we shall introduce a general theory of irreducible NIM representation over  fusion ring. We give the minimum upper bound of rank of NIM representation and the general classification methods of irreducible NIM representations over the near group fusion rings. Moreover we give explicitly the construction of irreducible NIM represen-tations over some near group fusion rings.