Chapter

Commutative Algebra

pp 293-305

Date:

Integral Closure of Rings of Integer-Valued Polynomials on Algebras

  • Giulio PeruginelliAffiliated withInstitut für Analysis und Comput. Number Theory, Technische University Email author 
  • , Nicholas J. WernerAffiliated withThe Ohio State University-Newark

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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