\begin{definition}[Definition:Addition of Polynomials]
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 $q$.
\end{definition}