Null Polynomial is Additive Identity

Theorem

The set of polynomial forms has an additive identity.

This article, or a section of it, needs explaining. In particular: Context You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by explaining it. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Explain}} from the code.

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:

\(\ds f + N\)

\(=\)

\(\ds \sum_{k \mathop \in Z} \paren {a_k + 0_R} \mathbf X^k\)

Definition of Polynomial Addition

\(\ds \)

\(=\)

\(\ds \sum_{k \mathop \in Z} a_k \mathbf X^k\)

\(\ds \)

\(=\)

\(\ds f\)

Definition of Polynomial Addition

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

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

$\blacksquare$