What’s New About Integer-Valued Polynomials on a Subset?

Abstract

The “classical” ring of integer-valued polynomials is the ring

$$ Int(Z) = \{ f \in Q[X]|f(Z) \subseteq Z\} $$

of integer-valued polynomials on Z. It is certainly one of the most natural ex­amples of a non-Noetherian domain.(Most rings studied in Commutative Algebra are Noetherian and so are the rings derived from a Noetherian ring by the classical algebraic constructions, such as localization, quotient, polynomials or power series in one indetermi­nate. To produce non-Noetherian rings one is led to consider ad hoc constructions, usually involving infinite extensions or the addition of infinitely many indeterminates, or else, to consider rings of functions as, for instance, the ring of entire functions.)