Definition:Monoid Ring

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.

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 $R$.

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.