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Integral Closure of Rings of Integer-Valued Polynomials on Algebras

Abstract

Let D be an integrally closed domain with quotient field K. Let A be a torsion-free D-algebra that is finitely generated as a D-module. For every a in A we consider its minimal polynomial μ a (X) ∈ D[X], i.e. the monic polynomial of least degree such that μ a (a) = 0. The ring Int K (A) consists of polynomials in K[X] that send elements of A back to A under evaluation. If D has finite residue rings, we show that the integral closure of Int K (A) is the ring of polynomials in K[X] which map the roots in an algebraic closure of K of all the μ a (X), aA, into elements that are integral over D. The result is obtained by identifying A with a D-subalgebra of the matrix algebra M n (K) for some n and then considering polynomials which map a matrix to a matrix integral over D. We also obtain information about polynomially dense subsets of these rings of polynomials.

Keywords

  • Integer-valued polynomial
  • Matrix
  • Triangular matrix
  • Integral closure
  • Pullback
  • Polynomially dense

MSC(2010) classification:

  • []13B25
  • 13B22
  • 11C20

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Acknowledgements

The authors wish to thank the referee for his/her suggestions. The first author wishes to thank Daniel Smertnig for useful discussions during the preparation of this paper about integrality in noncommutative settings. The same author was supported by the Austrian Science Foundation (FWF), Project Number P23245-N18.

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Correspondence to Giulio Peruginelli .

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Peruginelli, G., Werner, N.J. (2014). Integral Closure of Rings of Integer-Valued Polynomials on Algebras. In: Fontana, M., Frisch, S., Glaz, S. (eds) Commutative Algebra. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-0925-4_17

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