Polynomial Functions form Submodule of All Functions

Theorem

Let $K$ be a commutative ring with unity.

Let $K^K$ be the $K$-module mappings $f: K \to K$.

Let $\map P K \subseteq K^K$ be the set of all polynomial functions on $K$.

Then $\map P K$ is a $K$-submodule of $K^K$.

Proof

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Sources

• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {V}$: Vector Spaces: $\S 27$. Subspaces and Bases: Example $27.4$