Category:Definitions/Monoid Rings

This category contains definitions related to Monoid Rings.
Related results can be found in Category:Monoid Rings.

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.

This article, or a section of it, needs explaining. In particular: 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. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by explaining it. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Explain}} from the code.

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 $R$.