Several Anzahl theorems of matrices over Galois rings and their applications

Let R denote the Galois ring of characteristic p s and cardinality p s h . 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 GL m ( R ) × GL n ( 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.