Editing Group ring (section)

{{Short description|The set of finitely supported functions from a group to a ring}}
{{about|the algebraic group ring of a group|the case of a topological group|group algebra of a topological group}}

In [[algebra]], a '''group ring''' is a [[free module]] and at the same time a [[Ring (mathematics)|ring]], constructed in a natural way from any given ring and any given [[Group (mathematics)|group]]. As a free module, its ring of scalars is the given ring, and its basis is the set of elements of the given group. As a ring, its addition law is that of the free module and its multiplication extends "by linearity" the given group law on the basis. Less formally, a group ring is a generalization of a given group, by attaching to each element of the group a "weighting factor" from a given ring.

If the ring is commutative then the group ring is also referred to as a '''group algebra''', for it is indeed an [[Algebra over a ring|algebra]] over the given ring. A group algebra over a field has a further structure of a [[Hopf algebra]]; in this case, it is thus called a [[group Hopf algebra]].

The apparatus of group rings is especially useful in the theory of [[group representation]]s.