\begin{definition}[Definition:Polynomial Ring/Monoid Ring on Natural Numbers]
Let $R$ be a commutative ring with unity.
Let $\N$ denote the additive monoid of natural numbers.
Let $R \left[{\N}\right]$ be the monoid ring of $\N$ over $R$.
The '''polynomial ring over $R$''' is the ordered triple $\left({R \left[{\N}\right], \iota, X}\right)$ where:
:$X \in R \left[{\N}\right]$ is the standard basis element associated to $1\in \N$.
:$\iota : R \to R \left[{\N}\right]$ is the canonical mapping.
\end{definition}