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.

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:

    It’s important to remember that feminism is no longer a group of organizations or leaders. It’s the expectations that parents have for their daughters, and their sons, too. It’s the way we talk about and treat one another. It’s who makes the money and who makes the compromises and who makes the dinner. It’s a state of mind. It’s the way we live now.
    Anna Quindlen (20th century)

    I was exceedingly interested by this phenomenon, and already felt paid for my journey. It could hardly have thrilled me more if it had taken the form of letters, or of the human face. If I had met with this ring of light while groping in this forest alone, away from any fire, I should have been still more surprised. I little thought that there was such a light shining in the darkness of the wilderness for me.
    Henry David Thoreau (1817–1862)