Ring of Polynomial Forms is not necessarily Isomorphic to Ring of Polynomial Functions

Theorem
Let $D$ be an integral domain.

Let $D \sqbrk X$ be the ring of polynomial forms in $X$ over $D$.

Let $\map P D$ be the ring of polynomial functions over $D$.

Then it is not necessarily the case that $D \sqbrk X$ is isomorphic with $\map P D$.

Proof
Note that we have

Also see

 * Epimorphism from Polynomial Forms to Polynomial Functions, where it is shown that there exists an epimorphism from $D \sqbrk X$ to $\map P D$.