Definition:Polynomial Ring/Monoid Ring on Natural Numbers

Definition
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.

Also see

 * Equivalence of Definitions of Polynomial Ring