Integer-valued polynomials over structural matrix rings

Abstract

We study integer-valued polynomials and null polynomials over structural matrix rings, that is, rings whose elements are matrices in which an entry may be nonzero only when its row index precedes its column index in a fixed preorder. We prove that the integer-valued polynomials over a structural matrix ring with entries in an integral domain form a ring, and that the null polynomials over a structural matrix ring with entries in an arbitrary commutative ring form a two-sided ideal. In both settings, we give a characterization of the corresponding polynomials with matrix coefficients in terms of scalar-coefficient polynomials. These results extend corresponding theorems for full matrix rings and upper triangular matrix rings.