Integer-valued skew polynomials

For a commutative integral domain D with field of fractions K, the ring of integer-valued polynomials on D is Int(D) = {f is an element of K[x] | f(a) is an element of Dfor all a is an element of D}. In this paper, we extend this construction to skew polynomial rings. Given an automorphism sigma of K, the skew polynomial ring K[x; sigma] consists of polynomials with coefficients from K, and with multiplication given by xa = sigma(a)x for all a is an element of K. We define Int(D; sigma) = {f is an element of K[x; sigma] | f(a) is an element of Dfor alla is an element of D}, which is the set of integer-valued skew polynomials on D. When sigma is not the identity, K[x; sigma] is noncommutative and evaluation behaves differently than it does for ordinary polynomials. Nevertheless, we are able to prove that Int(D; sigma) has a ring structure in many cases. We show how to produce elements of Int(D; sigma) and investigate its properties regarding localization and Noetherian conditions. Particular attention is paid to the case where D is a discrete valuation ring with finite residue field.