Several Anzahl theorems of matrices over Galois rings and their applications

Let R denote the Galois ring of characteristic ps and cardinality psh. In this paper, we determine the Smith normal forms of matrices over R, compute the number of the orbits of m×n matrices under the group GLm(R)×GLn(R) and the length of each orbit. Moreover, we discuss their applications to association schemes, systems of linear equations, generalized invertible matrices, idempotent matrices and involutory matrices.