Definition:Ring of Polynomial Functions

Theorem
Let $(R,+,\circ)$ be a commutative ring with unity.

Let $R \left[\left\{{X_j: j \in J}\right\}\right]$ be the ring of polynomial forms over $R$ in the indeterminates $\left\{{X_j: j \in J}\right\}$.

Let $R^J$ be the free module on $J$.

Let $A$ be the set of all polynomial functions $R^J \to R$ associated elements of $R \left[\left\{{X_j: j \in J}\right\}\right]$.

Then the operations $+$ and $\circ$ on $R$ induce operations on $A$.

We denote these operations by the same symbols:


 * $\forall x \in R^J: \left({f + g}\right) \left({x}\right) = f \left({x}\right) + g \left({x}\right)$


 * $\forall x \in R^J: \left({f \circ g}\right) \left({x}\right) = f \left({x}\right)\circ g \left({x}\right)$

Then $\left({A, +, \circ}\right)$ is a commutative ring with unity.

Proof
First we check that $A$ is closed under multiplication and addition.

Let $Z$ be the set of all multiindices indexed by $J$.

Let $\displaystyle f=\sum_{k\in Z}a_k\mathbf X^k,\ \displaystyle g=\sum_{k\in Z}b_k\mathbf X^k \in R \left[\left\{{X_j: j \in J}\right\}\right]$.

Under the evaluation homomorphism, $f$ and $g$ map to


 * $\displaystyle A \owns \hat f : \forall x\in R^J: \hat f \left({x}\right) = \sum_{k \in Z} a_k x^k$


 * $\displaystyle A \owns \hat g : \forall x\in R^J: \hat g \left({x}\right) = \sum_{k \in Z} b_k x^k$

Then the induced sum of $\hat f$ and $\hat g$ is

Thus polynomial functions are closed under addition.

The induced product of $\hat f$ and $\hat g$ is

Thus polynomial functions are closed under multiplication.

Finally, we invoke Induced Ring, which shows that $\left({ A, +, \circ }\right)$ is a commutative ring with unity.

Note
By Equality of Polynomials, when $R$ is an infinite field, the ring of polynomial functions is isomorphic to the ring of polynomial forms. In this case, it is customary to write $R \left[\left\{{X_j: j \in J}\right\}\right]$ for the ring of polynomial functions also.