Definition:Addition of Polynomials

Definition
Let $\struct {R, +, \circ}$ be a ring.

Let $\struct {S, +, \circ}$ be a subring of $R$.

For arbitrary $x \in R$, let $S \sqbrk x$ be the set of polynomials in $x$ over $S$.

Let $p, q \in S \sqbrk x$ be polynomials in $x$ over $S$:
 * $\ds p = \sum_{k \mathop = 0}^m a_k \circ x^k$
 * $\ds q = \sum_{k \mathop = 0}^n b_k \circ x^k$

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:
 * $\ds p + q := \sum_{k \mathop = 0}^{\map \max {m, n} } \paren {a_k + b_k} x^k$

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 $q$.