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
Famous quotes containing the words group and/or ring:
“The virtue of dress rehearsals is that they are a free show for a select group of artists and friends of the author, and where for one unique evening the audience is almost expurgated of idiots.”
—Alfred Jarry (1873–1907)
“This is the gospel of labour, ring it, ye bells of the kirk!
The Lord of Love came down from above, to live with the men who work.
This is the rose that He planted, here in the thorn-curst soil:
Heaven is blest with perfect rest, but the blessing of Earth is toil.”
—Henry Van Dyke (1852–1933)