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=DdD 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.