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

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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. We compute a 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. We discuss two applications. At first, 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 of all integer-valued polynomials on A.