Definition:Polynomial Addition

Definition
Let $\left({R, +, \circ}\right)$ be a ring.

Let $\left({S, +, \circ}\right)$ be a subring of $R$.

For arbitrary $x \in R$, let $S \left[{x}\right]$ be the set $S \left[{x}\right]$ be the set of polynomials in $x$ over $S$.

Let $p, q \in S \left[{x}\right]$ be polynomials in $x$ over $S$:
 * $\displaystyle p = \sum_{k \mathop = 0}^m a_k \circ x^k$
 * $\displaystyle q = \sum_{k \mathop = 0}^n b_k \circ x^k$

where:
 * $(1): \quad a_k, b_k \in S$ for all $k$
 * $(2): \quad m, n \in \Z_{\ge 0}$.

The operation polynomial addition is defined as:
 * $\displaystyle p + q := \sum_{k \mathop = 0}^{\max \left({m, n}\right)} \left({a_k + b_k}\right) x^k$

where:
 * $\forall k \in \Z: k > m \implies a_k = 0$
 * $\forall k \in \Z: k > n \implies b_k = 0$

The expression $p + q$ is known as the sum of $p$ and $q$.