Integer-valued polynomials on algebras

TL;DR

This paper studies integer-valued polynomials on algebras over a domain, analyzing their properties, continuity, and algebraic structure, including spectra and Krull dimension, especially for matrix algebras.

Contribution

It extends the theory of integer-valued polynomials to general algebras, providing new results on continuity and algebraic structure, including for matrix algebras.

Findings

I-adic continuity of integer-valued polynomials established

Spectrum and Krull dimension determined for certain domains

Results applied to matrix algebras over domains

Abstract

Let D be a domain with quotient field K and A a D-algebra. We call a polynomial with coefficients in K that maps every element of A to an element of A "integer-valued on A". For commutative A we also consider integer-valued polynomials in several variables. For an arbitrary domain D and I an arbitrary ideal of D we show I-adic continuity of integer-valued polynomials on A. For Noetherian one-dimensional D, we determine the spectrum and Krull dimension of the ring Int_D(A) of integer-valued polynomials on A. We do the same for the ring of polynomials with coefficients in M_n(K), the K-algebra of n x n matrices, that map every matrix in M_n(D) to a matrix in M_n(D).