Definition:Monoid Ring

Definition
Let $R$ be a ring.

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

Let $R^{(G)}$ denote the free $R$-module indexed by $G$.

We define the operation $\circ$ on $R^{(G)}$ as a bilinear map, which we define on the canonical basis $\left\{{ e_g }~\middle\vert~{ g\in G }\right\}$ of $R^{(G)}$ by the law:


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

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

By construction, multiplication is distributive over addition.

It can be verified that the multiplication is indeed associative.

Properties

 * 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.


 * If $G$ is a group, $R[G]$ is a group ring.