\begin{definition}[Definition:Polynomial Function]
Let $R$ be a commutative ring with unity.
Let the mapping $p: R \to R$ be defined such that there exists a sequence:
: $\left \langle {\alpha_k} \right \rangle_{k \in \left[{0 \,.\,.\, n}\right]}$
of elements of $R$ such that:
:$\displaystyle p = \sum_{k \mathop = 0}^n \alpha_k {\operatorname{id}_R}^k$
where $\operatorname{id}_R$ is the identity mapping on $R$.
Then $p$ is known as a '''polynomial function on $R$ in one variable'''.
{{explain|$\operatorname{id}^k$ is not defined. What is wrong with a placeholder $x$?}}
\end{definition}