Definition:Monoid Ring

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Let $R$ be a ring with unity.

Let $\left({G, *}\right)$ be a monoid.

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

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

By Multilinear Mapping from Free Modules is Determined by Bases, there exists a unique bilinear map:

$\circ: R^{\left({G}\right)} \times R^{\left({G}\right)} \to R^{\left({G}\right)}$

which satisfies:

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

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

Canonical Mapping

Let $e_1$ be the canonical basis element.

The canonical mapping to $R \left[{G}\right]$ is the mapping $R \to R \left[{G}\right]$ which sends $r$ to $r * e_1$.

Also see