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:
“If the Russians have gone too far in subjecting the child and his peer group to conformity to a single set of values imposed by the adult society, perhaps we have reached the point of diminishing returns in allowing excessive autonomy and in failing to utilize the constructive potential of the peer group in developing social responsibility and consideration for others.”
—Urie Bronfenbrenner (b. 1917)
“The world,—this shadow of the soul, or other me, lies wide around. Its attractions are the keys which unlock my thoughts and make me acquainted with myself. I run eagerly into this resounding tumult. I grasp the hands of those next to me, and take my place in the ring to suffer and to work, taught by an instinct, that so shall the dumb abyss be vocal with speech.”
—Ralph Waldo Emerson (1803–1882)