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 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
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 .
Examples
1. Let , 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.e. C[G] is isomorphic to the ring C[]/.
Writing a different element s as , their sum is
and their product is
Notice that the identity element 1G of G induces a canonical embedding of the coefficient ring (in this case C) into C[G]; however strictly speaking the multiplicative identity element of C[G] is 1⋅1G where the first 1 comes from C and the second from G. The additive identity element is zero.
When G is a non-commutative group, one must be careful to preserve the order of the group elements (and not accidentally commute them) when multiplying the terms.
2. A different example is that of the Laurent polynomials over a ring R: these are nothing more or less than the group ring of the infinite cyclic group Z over R.
3. Let Q be the quaternion group with elements . Consider the group ring RQ, where R is the set of real numbers. An arbitrary element of this group ring is of the form
where is a real number.
Multiplication, as in any other group ring, is defined based on the group operation. For example,
Note that RQ is not the same as the skew field of quaternions over R. This is because the skew field of quaternions satisfies additional relations in the ring, such as , whereas in the group ring RQ, is not equal to . To be more specific, the group ring RQ has dimension 8 as a real vector space, while the skew field of quaternions has dimension 4 as a real vector space.
4. Another example of a non-abelian group ring is where is the symmetric group on 3 letters. This is not an integral domain since we have where the element is a transposition-a permutation which only swaps 1 and 2. Therefore the group ring need not be an integral domain even when the underlying ring is an integral domain.
Some basic properties
Using 1 to denote the multiplicative identity of the ring R, and denoting the group unit by 1G, the ring R[G] contains a subring isomorphic to R, and its group of invertible elements contains a subgroup isomorphic to G. For considering the indicator function of { 1G}, which is the vector f defined by
the set of all scalar multiples of f is a subring of R[G] isomorphic to R. And if we map each element s of G to the indicator function of { s}, which is the vector f defined by
the resulting mapping is an injective group homomorphism (with respect to multiplication, not addition, in R[G]).
If R and G are both commutative (i.e., R is commutative and G is an abelian group), R[G] is commutative.
If H is a subgroup of G, then R[H] is a subring of R[G]. Similarly, if S is a subring of R, S[G] is a subring of R[G].