\begin{definition}[Definition:Monoid Ring]
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.
{{explain|Before the above notation can be properly understood, the precise nature of the canonical basis needs to be expanded so as to make the operations completely explicit.}}
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 $R$'''.
\end{definition}