Interpolation domain

This article defines a property of integral domains, viz., a property that, given any integral domain, is either true or false for that.
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Definition

An integral domain $R$ is termed an interpolation domain if, for any nonnegative integer $n$ , there exist elements $(r_{0}, r_{1}, \dots, r_{n}) \in R^{n + 1}$ giving a bijection between the polynomials in $I n t (R)$ (the ring of integer-valued polynomials over $R$ ) of degree at most $n$ and the elements of $R^{n + 1}$ by:

$f \mapsto (f (r_{0}), f (r_{1}), \dots, f (r_{n}))$ .

In other words, for any tuple $(a_{0}, a_{1}, \dots, a_{n}) \in R^{n + 1}$ , there exists a unique $f \in I n t (R)$ such that $f (r_{i}) = a_{i}$ for every $0 \leq i \leq n$ .

Examples

Every field is an interpolation domain. This can be seen in many ways; one of them is the Lagrange interpolation formula.
The ring of rational integers is an interpolation domain. We can set $r_{i} = i$ for this ring. In other words, given any sequence of $n + 1$ integers, we can find an integer-valued polynomial sending $0, 1, \dots, n$ to the elements of that sequence. For full proof, refer: Ring of rational integers is an interpolation domain