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
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—Vladimir Nabokov (1899–1977)
“Generally, about all perception, we can say that a sense is what has the power of receiving into itself the sensible forms of things without the matter, in the way in which a piece of wax takes on the impress of a signet ring without the iron or gold.”
—Aristotle (384–323 B.C.)