Definition:Polynomial Function/Ring/Definition 2

Definition
Let $R$ be a commutative ring with unity.

Let $S \subset R$ be a subset. Let $R \sqbrk X$ be the polynomial ring in one variable over $R$.

Let $R^S$ be the ring of mappings from $S$ to $R$.

Let $\iota \in R^S$ denote the inclusion $S \hookrightarrow R$.

A polynomial function on $S$ is a mapping $f : S \to R$ which is in the image of the evaluation homomorphism $R \sqbrk X \to R^S$ at $\iota$.

Also see

 * Equivalence of Definitions of Polynomial Function on Subset of Ring