Definition:Monoid Ring

Definition

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.

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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 $R$.

Canonical Mapping

Let $e_1$ be the canonical basis element.

The canonical mapping to $R \sqbrk G$ is the mapping $R \to R \sqbrk G$ which sends $r$ to $r * e_1$.

Also see

• Monoid Ring is Ring, where it is shown that $R \sqbrk G$ is a ring.
• Definition:Big Monoid Ring
• Definition:Group Ring
• Universal Property of Monoid 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.

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