Polynomials Closed under Addition

Theorem
Let $\displaystyle f = \sum_{k\in Z}a_k\mathbf X^k$, $\displaystyle g = \sum_{k\in Z}b_k\mathbf X^k$ be polynomials in the indeterminates $\left\{{X_j: j \in J}\right\}$ over the ring $R$.

Define the sum


 * $\displaystyle f \oplus g = \sum_{k \in Z}(a_k + b_k) \mathbf X^k$

Then $f \oplus g$ is a polynomial.

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 $k \in Z$, $a_k + b_k \neq 0$

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.