Definition:Ring of Polynomial Functions

Definition
Let $\left({R, +, \circ}\right)$ 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 $R^J \to R$.

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

We denote these operations by the same symbols:


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


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

The ring of polynomial functions is the resulting algebraic structure.

Also see

 * Definition:Polynomial Function
 * Ring of Polynomial Functions is Commutative Ring with Unity
 * Equality of Polynomials, where it is shown that when $R$ is an infinite field, the ring of polynomial functions is isomorphic to the ring of polynomial forms.


 * In such a case, it is customary to write $R \left[{\left\{{X_j: j \in J}\right\}}\right]$ for the ring of polynomial functions also.