Integral Closure of Rings of Integer-Valued Polynomials on Algebras
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 μa(X) ∈ D[X], i.e. the monic polynomial of least degree such that μa(a) = 0. The ring IntK(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 IntK(A) is the ring of polynomials in K[X] which map the roots in an algebraic closure of K of all the μa(X), a ∈ A, into elements that are integral over D. The result is obtained by identifying A with a D-subalgebra of the matrix algebra Mn(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.
KeywordsInteger-valued polynomial Matrix Triangular matrix Integral closure Pullback Polynomially dense
MSC(2010) classification:13B25 13B22 11C20
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