Group ring

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.

Additional structure
We can equip the group ring with some additional structure, namely a coalgebra structure and an antipode map, thus turning it into a Hopf algebra.

Related notions

 * Skew group ring is an analogous notion where we use an action of the group on the ring.