# Definition:Monoid Ring/Canonical Mapping

## Definition

Let $R$ be a ring with unity.

Let $\left({G, *}\right)$ be a monoid with identity element $1$.

Let $R \left[{G}\right]$ be the monoid ring of $G$ over $R$.

Let $e_1$ be the canonical basis element.

The canonical mapping to $R \left[{G}\right]$ is the mapping $R \to R \left[{G}\right]$ which sends $r$ to $r * e_1$.