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 the set of elements of 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 ring is commutative then the group ring is also referred to as a group algebra, for it is indeed an algebra over the given ring. A group algebra over a field has a further structure of a Hopf algebra; in this case, it is thus called a group Hopf algebra. The apparatus of group rings is especially useful in the theory of group representations.
Definition Let G {\displaystyle G} be a group, written multiplicatively, and let R {\displaystyle R} be a ring. The group ring of G {\displaystyle G} over R {\displaystyle R} , which we will denote by R [ G ] {\displaystyle R[G]} , or simply R G {\displaystyle RG} , is the set of mappings f : G → R {\displaystyle f\colon G\to R} of finite support ( f ( g ) {\displaystyle f(g)} is nonzero for only finitely many elements g {\displaystyle g} ), where the module scalar product α f {\displaystyle \alpha f} of a scalar α {\displaystyle \alpha } in R {\displaystyle R} and a mapping f {\displaystyle f} is defined as the mapping x ↦ α ⋅ f ( x ) {\displaystyle x\mapsto \alpha \cdot f(x)} , and the module group sum of two mappings f {\displaystyle f} and g {\displaystyle g} is defined as the mapping x ↦ f ( x ) + g ( x ) {\displaystyle x\mapsto f(x)+g(x)} . To turn the additive group R [ G ] {\displaystyle R[G]} into a ring, we define the product of f {\displaystyle f} and g {\displaystyle g} to be the mapping
x ↦ ∑ u v = x f ( u ) g ( v ) = ∑ u ∈ G f ( u ) g ( u − 1 x ) . {\displaystyle x\mapsto \sum _{uv=x}f(u)g(v)=\sum _{u\in G}f(u)g(u^{-1}x).}
The summation is legitimate because f {\displaystyle f} and g {\displaystyle g} are of finite support, and the ring axioms are readily verified. Some variations in the notation and terminology are in use. In particular, the mappings such as f : G → R {\displaystyle f:G\to R} are sometimes written as what are called "formal linear combinations of elements of G {\displaystyle G} with coefficients in R {\displaystyle R}
":
∑ g ∈ G f ( g ) g , {\displaystyle \sum _{g\in G}f(g)g,}
or simply
∑ g ∈ G f g g . {\displaystyle \sum _{g\in G}f_{g}g.}
Note that if the ring R {\displaystyle R} is in fact a field, then the module structure of the group ring R G {\displaystyle RG} is in fact a vector space over R {\displaystyle R} .
Examples 1. Let G = C3, the cyclic group of order 3, with generator a {\displaystyle a} and identity element 1G. An element r of C[G] can be written as
r = z 0 1 G + z 1 a + z 2 a 2 {\displaystyle r=z_{0}1_{G}+z_{1}a+z_{2}a^{2}\,}
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