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 $M_n(D)$ be the ring of $n\times n$ matrices with entries in $D$. The ring of integer-valued polynomials on the matrix ring $M_n(D)$, denoted ${\mathrm{Int}}_K(M_n(D))$, consists of those polynomials in $K[x]$ that map matrices in $M_n(D)$ back to $M_n(D)$ under evaluation. It has been known for some time that ${\mathrm{Int}}_{\mathbb{Q}}(M_n(\mathbb{Z}))$ is not integrally closed. However, it was only recently that an example of a polynomial in the integral closure of ${\mathrm{Int}}_{\mathbb{Q}}(M_n(\mathbb{Z}))$ 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 ${\mathrm{Int}}_K(M_n(D))$ but are not in the ring itself, where $D$ is a Dedekind domain with finite residue fields and $n\ge 2$ is arbitrary. We also show how our general example is related to $P$-sequences for ${\mathrm{Int}}_K(M_n(D))$ and its integral closure in the case where $D$ is a discrete valuation ring.

arXiv commentfinal version, to appear in J. Algebra (2016); comments are welcome!

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• Integral closure of rings of integer-valued polynomials on algebras
• Integer-valued polynomials over matrices and divided differences
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Cited by in corpus (4)

• Non-triviality Conditions for Integer-valued Polynomial Rings on Algebras
• Decomposition of integer-valued polynomial algebras
• Galois structure on integral valued polynomials
• A Survey on Fixed Divisors