\begin{definition}[Definition:Polynomial in Ring Element/Definition 2]
Let $R$ be a commutative ring.
Let $S$ be a subring with unity of $R$.
Let $x\in R$.
Let $S[X]$ be the polynomial ring in one variable over $S$.
A '''polynomial in $x$ over $S$''' is an element that is in the image of the evaluation homomorphism $S[X]\to R$ at $x$.
\end{definition}