# Definition:Monoid Ring

Jump to navigation
Jump to search

## Definition

Let $R$ be a ring with unity.

Let $\struct {G, *}$ be a monoid.

Let $R^{\paren G}$ be the free $R$-module on $G$.

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

This article, or a section of it, needs explaining.In particular: Before the above notation can be properly understood, the precise nature of the canonical basis needs to be expanded so as to make the operations completely explicit.You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by explaining it.To discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{Explain}}` from the code. |

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

- $\circ: R^{\paren G} \times R^{\paren G} \to R^{\paren G}$

which satisfies:

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

Then $R \sqbrk G = \struct {R^{\paren G}, +, \circ}$ is called the **monoid ring of $G$ over $R$**.

### Canonical Mapping

Let $e_1$ be the canonical basis element.

The **canonical mapping to $R \sqbrk G$** is the mapping $R \to R \sqbrk G$ which sends $r$ to $r * e_1$.

## Also see

- Monoid Ring is Ring, where it is shown that $R \sqbrk G$ 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.

This article, or a section of it, needs explaining.You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by explaining it.To discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{Explain}}` from the code. |

## Sources

There are no source works cited for this page.Source citations are highly desirable, and mandatory for all definition pages.Definition pages whose content is wholly or partly unsourced are in danger of having such content deleted. To discuss this page in more detail, feel free to use the talk page. |