Properly integral polynomials over the ring of integer-valued polynomials on a matrix ring

Abstract

Let D be a domain with fraction field K, and let Mn(D) be the ring of n×n matrices with entries in D. The ring of integer-valued polynomials on the matrix ring Mn(D), denoted IntK(Mn(D)), consists of those polynomials in K[x] that map matrices in Mn(D) back to Mn(D) under evaluation. It has been known for some time that IntQ(Mn(Z)) is not integrally closed. However, it was only recently that an example of a polynomial in the integral closure of IntQ(Mn(Z)) but not in the ring itself appeared in the literature, and the published example is specific to the case n=2. In this paper, we give a construction that produces polynomials that are integral over IntK(Mn(D)) but are not in the ring itself, where D is a Dedekind domain with finite residue fields and n≥2 is arbitrary. We also show how our general example is related to P-sequences for IntK(Mn(D)) and its integral closure in the case where D is a discrete valuation ring.