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Concept
Group ring
Summary
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.
Let be a group, written multiplicatively, and let be a ring. The group ring of over , which we will denote by , or simply , is the set of mappings of finite support ( is nonzero for only finitely many elements ), where the module scalar product of a scalar in and a mapping is defined as the mapping , and the module group sum of two mappings and is defined as the mapping . To turn the additive group into a ring, we define the product of and to be the mapping
The summation is legitimate because and 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 are sometimes written as what are called "formal linear combinations of elements of , with coefficients in
":
or simply
where this doesn't cause confusion.
Note that if the ring is in fact a field , then the module structure of the group ring is in fact a vector space over .
- Let G = C3, the cyclic group of order 3, with generator and identity element 1G. An element r of C[G] can be written as where z0, z1 and z2 are in C, the complex numbers. This is the same thing as a polynomial ring in variable such that i.
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