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Given a square matrix $A$ with entries in a commutative ring $S$, the idealof $S[X]$ consisting of polynomials $f$ with $f(A) =0$ is called the null idealof $A$. Very little is known about null ideals of matrices over generalcommutative rings. We compute a generating set of the null ideal of a matrix incase $S = D/dD$ is the residue class ring of a principal ideal domain $D$modulo $d\in D$. We discuss two applications. At first, we compute adecomposition of the $S$-module $S[A]$ into cyclic $S$-modules and explain thestrong relationship between this decomposition and the determined generatingset of the null ideal of $A$. And finally, we give a rather explicitdescription of the ring \IntA of all integer-valued polynomials on $A$.

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