Null ideals of matrices over residue class rings of principal ideal domains

Given a square matrix A with entries in a commutative ring S,
the ideal of S[X] consisting of polynomials f with f (A) = 0
is called the null ideal of A. Very little is known about null
ideals of matrices over general commutative rings. First, we
determine a certain generating set of the null ideal of a matrix
in case S = D/dD is the residue class ring of a principal
ideal domain D modulo d ∈ D. After that we discuss two
applications. We compute a decomposition of the S-module
S[A] into cyclic S-modules and explain the strong relationship
between this decomposition and the determined generating set
of the null ideal of A. And finally, we give a rather explicit
description of the ring Int(A, Mn(D)) of all integer-valued
polynomials on A.