The ring constructed (notation *K*[*G*], or sometimes just *KG*) can be described as the vector space over *K* with basis the elements g of *G*, and ring multiplication the group operation in *G* extended by bilinearity to the whole space. That is, g_{1}g_{2} = g_{3} as an equation in *G* still holds true in *K*[*G*], and the whole structure of *K*[*G*] as an algebra over K follows when we apply the distributive law and *K*-linearity.

It is then true that a module *M* for *K*[*G*] is just what is normally meant as a *linear representation* of *G* over the field *K*. There is no particular reason to limit *K* to be a field here; but the classical results that were obtained first when *K* is the complex number field and *G* a finite group justify close attention to this case. It was shown that *K*[*G*] is a semisimple ring, under those conditions, with profound implications for the representations of finite groups.

When *G* is a finite abelian group, the group ring is commutative, and its structuer easy to express in terms of roots of unity. When *K* 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 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 an infinite cyclic group.

There is a neat characterisation from category theory of the group ring construction as adjoint to the functor taking a *K*-algebra to its group of units.

For the purposes of functional analysis, and in particular of harmonic analysis, one wishes to carry over the group ring construction to topological groups *G*. In this case the group ring should consist of enough functions on *G*, with product the convolution: in the algebraic case one already sees that if group ring elements are written as functions F(g) of group elements, with values in *K*, then the ring product is a type of convolution.

In practice there are various candidates for *the***complex group algebra** of *G*. One can take the algebra of continuous complex functions on *G* that are zero outside a compact set: then the convolution product will present no convergence difficulties. In case *G* is compact this is just C(*G*), the ring of all continuous functions on *G* with the convolution product. In this way one will get a C-star algebra.

In general the need to apply Fubini's theorem leads to the use of L^{1}(*G*), L^{2}(*G*), and their intersection. That is, to deal with the analytical difficulties one needs a range of algebras, instead of just one.