Definition:Monoid Ring

Definition
Let $R$ be a ring.

Let $M$ be a monoid.

Let $R^{(M)}$ denote the $R$-module that is the direct sum of copies of $R$ indexed by $M$.

We define a product on $R^{(M)}$ as a bilinear map, which we define on the basis $\{e_m\mid m\in M\}$ by the law
 * $e_m\cdot e_n=e_{mn}$

This is called the polynomial ring of $R$ over $M$, denoted $R[M]$.

By construction, multiplication is distributive over addition.

It can be verified that the multiplication is indeed associative.

Properties
If $R$ is commutative and $M$ is commutative, then $R[M]$ is commutative.

Examples
If $M=\N$, we get the ordinary ring of polynomials in one variable.

If $M=\N^n$, we get the ring of polynomials in $n$ variables.