Group ring

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[G] (or sometimes just RG) such that the multiplication in R[G] is induced by the multiplication in G.

R[G] can be described as the free module (if R is a field, this is just a vector space) with basis the elements g of G, and ring multiplication the group operation in G extended by bilinearity to the whole space. That is, g1g2 = g3 as an equation in G still holds true in R[G], and the whole structure of R[G] 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].

It is then true that a module M over R[G] is 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; but the classical results that were obtained first when R is the complex number field and G a finite group justify close attention to this case. It was shown that R[G] is a semisimple 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] 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.

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.

There is a neat characterisation 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.

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


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