\begin{definition}[Definition:Polynomial in Ring Element/Definition 1]
Let $R$ be a commutative ring.
Let $S$ be a subring with unity of $R$.
Let $x \in R$.
A '''polynomial in $x$ over $S$''' is an element $y \in R$ for which there exist:
:a natural number $n \in \N$
:$a_0, \ldots, a_n \in S$
such that:
:$y = \ds \sum_{k \mathop = 0}^n a_k x^k$
where:
:$\ds \sum$ denotes indexed summation
:$x^k$ denotes the $k$th power of $x$
\end{definition}