# Group Ring

In algebra, a group ring is a free module and at the same time a ring, constructed in a natural way from any given ring and any given group. As a free module, its ring of scalars is the given ring, and its basis is one-to-one with 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 given ring is commutative, a group ring is also referred to as a group algebra, for it is indeed an algebra over the given ring.

The apparatus of group rings is especially useful in the theory of group representations.

### Other articles related to "group ring, group, ring, groups":

Augmentation Ideal
... In algebra, an augmentation ideal is an ideal that can be defined in any group ring ... If G is a group and R a commutative ring, there is a ring homomorphism, called the augmentation map, from the group ring to R, defined by taking a sum to Here ri is an element ... The sums are finite, by definition of the group ring ...