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":
... 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 ...
... 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 (18031882)
“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 (18881975)