# Definition:Monoid Ring

## Definition

Let $R$ be a ring with unity.

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

Let $R^{\left({G}\right)}$ be the free $R$-module on $G$.

Let $\left\{ {e_g: g \in G}\right\}$ be its canonical basis.

By Multilinear Mapping from Free Modules is Determined by Bases, there exists a unique bilinear map:

- $\circ: R^{\left({G}\right)} \times R^{\left({G}\right)} \to R^{\left({G}\right)}$

which satisfies:

- $e_g \circ e_h = e_{g \mathop * h}$

Then $R \left[{G}\right] = \left({R^{\left({G}\right)}, +, \circ}\right)$ is called the **monoid ring of $G$ over $R$**.

### Canonical Mapping

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 $r * e_1$.

## Also see

- Monoid Ring is Ring, where it is shown that $R \left[{G}\right]$ is a ring.
- Definition:Big Monoid Ring
- Definition:Group Ring
- Universal Property of Monoid Ring
- Monoid Ring of Commutative Monoid over Commutative Ring is Commutative

## Examples

- If $G = \N$, we get the ordinary ring of polynomials in one variable.

- If $G = \N^n$, we get the ring of polynomials in $n$ variables.