J-ideals of matrices over PIDs

Given a square matrix B over a commutative ring R and an ideal J of R, the J-ideal is the ideal consisting of the polynomials f in R[X] such that all entries of f(B) are in J. J-ideals arise naturally in the study of integer-valued polynomials on a single matrix. If the underlying ring is a principal ideal domain it suffices to determine a finite number of polynomials in order to describe all J-ideals of B. In this talk we discuss new results concerning certain generating sets of J-ideals