Equivalence of Definitions of Polynomial Function on Subset of Ring

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Theorem

Let $R$ be a commutative ring with unity.

Let $S \subset R$ be a subset.

The following definitions of the concept of polynomial function are equivalent:

Definition 1

A polynomial function on $S$ is a mapping $f : S \to R$ for which there exist:

a natural number $n \in \N$
$a_0, \ldots, a_n \in R$

such that for all $x\in S$:

$\map f x = \displaystyle \sum_{k \mathop = 0}^n a_k x^k$

where $\sum$ denotes indexed summation.


Definition 2

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 $\iota$.


Outline of proof

This follows by interpreting both definitions as $f$ being a polynomial in the element $\iota$.


Proof

1 implies 2

Let $\map P X \in R \sqbrk X$ be the polynomial:

$P = \displaystyle \sum_{k \mathop = 0}^n a_k \cdot X^k$

where $\sum$ denotes indexed summation.

We show that $\map P \iota = f$.



2 implies 1

Let $P \in R \sqbrk X$ be a polynomial and $f = \map P \iota \in R^S$.

By Polynomial is Linear Combination of Monomials, there exist:

  • $n \in \N$
  • $a_0, \ldots, a_n \in R$

such that $P = \displaystyle \sum_{k \mathop = 0}^n a_k \cdot X^k$ where $\sum$ denotes indexed summation.

Let $\operatorname{ev}_{\iota}$ denote the evaluation homomorphism at $\iota$.

Then $\map {\operatorname{ev}_{\iota} } P = f$.


We have:

\(\displaystyle f\) \(=\) \(\displaystyle \map {\operatorname{ev}_{\iota} } {\displaystyle \sum_{k \mathop = 0}^n a_k \cdot X^k}\)
\(\displaystyle \) \(=\) \(\displaystyle \displaystyle \sum_{k \mathop = 0}^n \map {\operatorname{ev}_{\iota} } {a_k \cdot X^k}\) Ring Homomorphism Preserves Indexed Summations
\(\displaystyle \) \(=\) \(\displaystyle \displaystyle \sum_{k \mathop = 0}^n a_k \cdot \map {\operatorname{ev}_{\iota} } {X^k}\) Definition of Polynomial Evaluation Homomorphism
\(\displaystyle \) \(=\) \(\displaystyle \displaystyle \sum_{k \mathop = 0}^n a_k \cdot \map {\operatorname{ev}_{\iota} } {X^k }\) Definition of Polynomial Evaluation Homomorphism