Integral Closure of Rings of Integer-Valued Polynomials on Algebras

  • Giulio Peruginelli
  • Nicholas J. Werner


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.


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

MSC(2010) classification:

[]13B25 13B22 11C20 



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.


  1. 1.
    N. Bourbaki, Commutative Algebra (Addison-Wesley Publishing Co., Reading, Mass, Hermann, Paris 1972)MATHGoogle Scholar
  2. 2.
    J.-P. Cahen, J.-L. Chabert, Integer-Valued Polynomials, vol. 48 (American Mathematical Society Surveys and Monographs, Providence, RI 1997)MATHGoogle Scholar
  3. 3.
    L.E. Dickson, Algebras and their Arithmetics (Dover Publications, New York 1960)MATHGoogle Scholar
  4. 4.
    S. Evrard, Y. Fares, K. Johnson, Integer valued polynomials on lower triangular integer matrices. Monats. für Math. 170(2), 147–160 (2013)CrossRefMATHMathSciNetGoogle Scholar
  5. 5.
    S. Frisch, Corrigendum to integer-valued polynomials on algebras. J. Algebra 373, 414–425 (2013), J. Algebra 412 282 (2014). DOI: 10.1016/j.jalgebra.2013.06.003.
  6. 6.
    S. Frisch, Integer-valued polynomials on algebras. J. Algebra 373, 414–425 (2013)CrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    R. Gilmer, W. Heinzer, D. Lantz, The Noetherian property in rings of integer-valued polynomials. Trans. Amer. Math. Soc. 338(1), 187–199 (1993)CrossRefMATHMathSciNetGoogle Scholar
  8. 8.
    T.Y. Lam, A First Course in Noncommutative Rings (Springer, New York, 1991)CrossRefMATHGoogle Scholar
  9. 9.
    K.A. Loper, N.J. Werner, Generalized rings of integer-valued polynomials. J. Number Theory 132(11), 2481–2490 (2012)CrossRefMATHMathSciNetGoogle Scholar
  10. 10.
    G. Peruginelli, Integral-valued polynomials over sets of algebraic integers of bounded degree. J. Number Theory 137, 241–255 (2014)CrossRefMATHMathSciNetGoogle Scholar
  11. 11.
    G. Peruginelli, Integer-valued polynomials over matrices and divided differences. Monatsh. Math. 173(4), 559–571 (2014)CrossRefMATHMathSciNetGoogle Scholar
  12. 12.
    J.P. Serre, A Course in Arithmetic (Springer, New York 1996)Google Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Institut für Analysis und Comput. Number TheoryTechnische UniversityGrazAustria
  2. 2.The Ohio State University-NewarkNewarkUSA

Personalised recommendations