Definition:Monoid Ring

Definition
Let $R$ be a ring with unity.

Let $(G, +)$ be a monoid.

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

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

By Multilinear Mapping from Free Modules is Determined by Bases, there exists a unique bilinear map
 * $\circ:R^{(G)}\times R^{(G)}\to R^{(G)}$

which satisfies
 * $e_g \circ e_h = e_{g + h}$.

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

In Monoid Ring is Ring, it is shown that $R[G]$ is a ring.

Also see

 * Definition:Group 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.