Polynomials Closed under Addition/Polynomial Forms

Theorem
Let:
 * $\ds f = \sum_{k \mathop \in Z} a_k \mathbf X^k$
 * $\ds g = \sum_{k \mathop \in Z} b_k \mathbf X^k$

be polynomials in the indeterminates $\set {X_j: j \in J}$ over the ring $R$.

Then the operation of polynomial addition on $f$ and $g$:

Define the sum:
 * $\ds f \oplus g = \sum_{k \mathop \in Z} \paren {a_k + b_k} \mathbf X^k$

Then $f \oplus g$ is a polynomial.

That is, the operation of polynomial addition is closed on the set of all polynomials on a given set of indeterminates $\set {X_j: j \in J}$.

Proof
It is immediate that $f \oplus g$ is a map from the free commutative monoid to $R$, so we need only prove  that $f \oplus g$ is nonzero on finitely many $\mathbf X^k$, $k \in Z$.

Suppose that for some $k \in Z$, $a_k + b_k \ne 0$

This forces at least one of $a_k$ and $b_k$ to be non-zero.

This can only be true for a finite number of terms because $f$ and $g$ are polynomials.

The result follows.