On rational matrix exact covering systems of ZnZn and its applications to Ramanujan's forty identities

For a nonsingular integer matrix B  , the set of cosets of the quotient module Zn/BZnZn/BZn forms an exact covering system (ECS) of ZnZn. In this paper, we use the Smith normal form to obtain another type of matrix ECS with rational entries which we call rational matrix ECS. Using rational matrix ECS of Z2Z2, we prove eight identities in Ramanujan's list of forty identities for the Rogers–Ramanujan functions, as well as some other identities.