Null Polynomial is Additive Identity

Theorem
The set of polynomials has an additive identity.

Proof
Let $(R, +, \circ)$ be a commutative ring with unity with zero $0_R$.

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

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

Let


 * $\displaystyle f = \sum_{k\in Z} a_k \mathbf X^k$

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

Let


 * $\displaystyle N = \sum_{k\in Z} 0_R \mathbf X^k$

be the null polynomial.

Then

Therefore, $N + f = f$ for all polynomials $f$.

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