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.

Let G be a group, written multiplicatively, and let R be a ring. The group ring of G over R, which we will denote by R[G] (or simply RG), is the set of mappings f : G → R of finite support, where the module scalar product αf of a scalar α in R and a vector (or mapping) f is defined as the vector ${\displaystyle x\mapsto \alpha \cdot f(x)}$, and the module group sum of two vectors f and g is defined as the vector ${\displaystyle x\mapsto f(x)+g(x)}$. To turn the additive group R[G] into a ring, we define the product of f and g to be the vector

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