Definition:Ring of Polynomial Functions

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Let $\struct {R, +, \circ}$ be a commutative ring with unity.

Let $R \sqbrk {\set {X_j: j \in J} }$ be the ring of polynomial forms over $R$ in the indeterminates $\set {X_j: j \in J}$.

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 pointwise operations on $A$.

We denote these operations by the same symbols:

$\forall x \in R^J: \map {\paren {f + g} } x = \map f x + \map g x$
$\forall x \in R^J: \map {\paren {f \circ g} } x = \map f x \circ \map g x$

The ring of polynomial functions is the resulting algebraic structure.

Also see

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