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.

Read more about Group Ring: Definition, Two Simple Examples, Some Basic Properties, Group Algebra Over A Finite Group, Group Rings Over An Infinite Group, Representations of A Group Ring, Filtration

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

... 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**...

... The Whitehead

**group**of the trivial

**group**is trivial ... Since the

**group ring**of the trivial

**group**is Z, we have to show that any matrix can be written as a product of elementary matrices times a diagonal matrix this follows easily from the fact that Z is a ... The Whitehead

**group**of a free abelian

**group**is trivial, a 1964 result of Bass, Heller and Swan ...

**Group Ring**- Filtration

... If a

**group**has a length function – for example, if there is a choice of generators and one takes the word metric, as in Coxeter

**groups**– then the

**group ring**becomes a filtered algebra ...

### Famous quotes containing the words ring and/or group:

“The life of man is a self-evolving circle, which, from a *ring* imperceptibly small, rushes on all sides outwards to new and larger circles, and that without end.”

—Ralph Waldo Emerson (1803–1882)

“It is not God that is worshipped but the *group* or authority that claims to speak in His name. Sin becomes disobedience to authority not violation of integrity.”

—Sarvepalli, Sir Radhakrishnan (1888–1975)