Given a group and a ring , the group ring or group algebra of over , denoted is defined as the following ring:
- Additively, it is a free -module with basis indexed by elements of
- The multiplication is defined as follows: the product of the basis element for and the basis element for is the basis element for . Multiplication on arbitrary elements is obtained by extending this rule -linearly.
Note that the group ring is an algebra over , with naturally sitting as the subring ( being the identity element).
Fix a base ring . We can then talk of the map sending any arbitrary group , to its group ring . This map is a functor from the category of groups to the category of -algebras. In other words, given any homomorphism of groups we geta corresponding homomorphism of algebras .
Further, this functor preserves injectivity and surjectivity: if the undiced map in injective. Similarly, if is a quotient of the induced map is surjective.
Representations of the group as modules over the group ring
Let be a -module with an action of on as -module automorphisms. Then, naturally acquires the structure of a - module.
- Any action of as automorphisms of an abelian group is equivalent to viewing the Abelian group as a module over (the group ring over the ring of integers).
- Any linear representation of over a field turns the vector space into a module.
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.
Further information: Group algebra as a Hopf algebra
- Skew group ring is an analogous notion where we use an action of the group on the ring.