In mathematics, the group ring is an algebraic construction that associates to a group G and a commutative ring with unity R an R-algebra R(or sometimes just RG) such that the multiplication in R[G is induced by the multiplication in G.

R^* can be described as the free module over R (if R is a field, this is just a vector space) generated by the elements g of G. The multiplication is the group operation in G extended by linearity and distributivity to the whole space.

The same notation is used for the elements of the group G and the basis elements of Rso that if in G we have g₁g₂ = g₃, then the same holds true in RG" target="_blank" >^*" target="_blank" >as an associative algebra over R follows when we apply the distributive law and R-linearity. The identity element of G serves as the 1 in R[G.

Definition

Let G be a group and R a ring. We first define the set RG to be one of the following:

• The set of all formal R-linear combinations of elements of G.
• The free R-module with basis G.
• The set of all functions f: G &rightarrow R with f(g) = 0 for all but finitely many g in G.

No matter which definition is used, we can write elements of RG in the form $\backslash sum\_\{g\; \backslash in\; G\}\; a\_g\; g$, with all but finitely many of the a_g, and an addition is defined on RG (by addition of formal linear combinations, addition in the module, or addition of functions, respectively). Multiplication of elements of RG is defined by setting

$(\backslash sum\_\{g\; \backslash in\; G\}\; a\_g\; g\; )(\; \backslash sum\_\{g\; \backslash in\; G\}\; b\_g\; g\; )\; \backslash \; =\; \backslash \; \backslash sum\_\{g,h\; \backslash in\; G\}\; (a\_g\; b\_h\; )\; gh$

If R has a unit element, this is the unique bilinear multiplication for which (1 g)(1 h) = (1 gh). In this case, G is commonly identified with the elements 1 g of RG. The identity element of G then serves as the 1 in R^*.

R is commonly a commutative ring with unit, or even a field.

Properties

If R and G are both commutative (i.e., R is commutative and G is an abelian group), RG is commutative.

If H is a subgroup of G, then RH is a subring of RG. Similarly, if S is a subring of R, SG is a subring of RG.

The case where G is finite

If G is finite, and R is the field of complex numbers, RG is a semisimple ring. This result, Maschke's theorem, allows us to understand RG as a ring of matrices with entries in R.

The case where G is infinite

Much less is known in the case where G is infinite, and this is an area of active research. The case where R is the field of complex numbers is probably the one best studied. In this case, Irving Kaplansky proved that if a and b are elements of CG with ab = 1, then ba = 1. Whether this is true if R is a field of finite characteristic remains unknown.

If G is torsion-free, it is conjectured that CG has no nontrivial idempotents or zero divisors; while this has been proven for special cases, such as the ones where G is abelian, elementary amenable, or free

An Example

Let G = Z₃, the cyclic group of 3 elements with generator a. Then an element of C^* is

z₁ + z₂a + z₃a²,

where z₁, z₂ and z₃ are in $\backslash mathbb\{C\}$, the complex numbers. If we take another element

w₁ + w₂a + w₃a²,

then their sum is

(z₁+w₁) + (z₂+w₂)a + (z₃+w₃)a²

and their multiplication is

(z₁ + z₂a + z₃a²) (w₁ + w₂a + w₃a²)

= (z₁w₁ + z₂w₃ + z₃w₂) + (z₁w₂ + z₂w₁ + z₃w₃)a + (z₁w₃ + z₂w₂ + z₃w₁)a².

In an example where G is a non-commutative group, we have to be careful to make the multiplication of the terms in the right order.

An example of a group ring of an infinite group is the ring of Laurent polynomials: this is exactly the group ring of the infinite cyclic group Z.

Representations

A module M over Ris then the same as a linear representation of G over the field R. There is no particular reason to limit R to be a field here. However, the classical results were obtained first when R is the complex number field and G is a finite group, so this case deserves close attention. It was shown that Rsemisimple ring, under those conditions, with profound implications for the representations of finite groups. More generally, whenever the characteristic of R does not divide the order of the finite group G, then R[G" target="_blank" >^* is semisimple (Maschke's theorem).

When G is a finite abelian group, the group ring is commutative, and its structure easy to express in terms of roots of unity. When R is a field of characteristic p, and the prime number p divides the order of the finite group G, then the group ring is not semisimple: it has a non-zero Jacobson radical, and this gives the corresponding subject of modular representation theory its own, deeper character.

Category Theory

There is an elegant characterization from category theory of the group ring construction as the left adjoint to the functor taking an associative R-algebra with one to its group of units.

Generalization

Group algebras are more general algebras which derive their multiplication from the multiplication in G.

Ring theory | Representation theory | Harmonic analysis | 群環