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 $R$.

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.