Group ring

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Definition

Given a group $G$ and a ring $R$, the group ring or group algebra of $G$ over $R$, denoted $R(G)$ is defined as the following ring:

• Additively, it is a free $R$-module with basis indexed by elements of $G$
• The multiplication is defined as follows: the product of the basis element for $g$ and the basis element for $h$ is the basis element for $gh$. Multiplication on arbitrary elements is obtained by extending this rule $R$-linearly.

Note that the group ring $R(G)$ is an algebra over $R$, with $R$ naturally sitting as the subring $Re$ ($e$ being the identity element).

Functoriality

Fix a base ring $R$. We can then talk of the map sending any arbitrary group $G$, to its group ring $R(G)$. This map is a functor from the category of groups to the category of $R$-algebras. In other words, given any homomorphism of groups $G \to H$ we geta corresponding homomorphism of algebras $R(G) \to R(H)$.

Further, this functor preserves injectivity and surjectivity: if $H \le G$ the undiced map $R(H) \to R(G)$ in injective. Similarly, if $H$ is a quotient of $G$ the induced map $R(G) \to R(H)$ is surjective.

Representations of the group as modules over the group ring

Let $M$ be a $R$-module with an action of $G$ on $M$ as $R$-module automorphisms. Then, $M$ naturally acquires the structure of a $R(G)$- module.

In particular:

• Any action of $G$ as automorphisms of an abelian group is equivalent to viewing the Abelian group as a module over $\Z(G)$ (the group ring over the ring of integers).
• Any linear representation of $G$ over a field $k$ turns the vector space into a $k(G)$ module.