Polynomials Addition is Associative

Theorem

Addition of polynomials is an associative operation.

Proof

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

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

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

$\displaystyle g = \sum_{k\in Z} b_k \mathbf X^k$

$\displaystyle h = \sum_{k\in Z} c_k \mathbf X^k$

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

Then

	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \left({ f + g }\right) + h\)	\(=\)	\(\displaystyle \sum_{k\in Z} \left({ \left({ a_k + b_k }\right) + c_k }\right) \mathbf X^k\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	applying the definition of polynomial addition twice
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \sum_{k\in Z} \left({ a_k + \left({ b_k + c_k }\right) }\right) \mathbf X^k\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	because addition in $R$ is associative
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle f + \left({ g + h }\right)\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	applying the definition of polynomial addition twice

$\blacksquare$