Null Polynomial is Additive Identity

Theorem
The set of polynomial forms has an additive identity.

Proof
Let $\struct {R, +, \circ}$ be a commutative ring with unity with zero $0_R$.

Let $\set {X_j: j \in J}$  be a set of indeterminates.

Let $Z$ be the set of all multiindices indexed by $\set {X_j: j \in J}$.

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

be an arbitrary polynomial form in the indeterminates $\set {X_j: j \in J}$ over $R$.

Let:
 * $\ds N = \sum_{k \mathop \in Z} 0_R \mathbf X^k$

be the null polynomial.

Then:

Therefore, $N + f = f$ for all polynomial forms $f$.

Therefore, $N$ is an additive identity for the set of polynomial forms.