Define the group ring \(R[G]\) and describe its basic structure and relation to representations.
Solution
Let \(R\) be a unital ring and \(G\) a group. The group ring \(R[G]\) consists of finite formal sums \[ \sum_{g\in G} r_g g, \qquad r_g\in R, \] with coefficientwise addition and multiplication determined by \[ (rg)(sh)=(rs)(gh) \] and distributivity.
As a left \(R\)-module, \(R[G]\) is free with basis \(G\). If \(R\) is commutative, then \(R[G]\) is an \(R\)-algebra. It is commutative exactly when \(R\) is commutative and \(G\) is abelian.
There is an augmentation homomorphism \[ \varepsilon:R[G]\to R, \qquad \sum r_g g\longmapsto \sum r_g, \] whose kernel is the augmentation ideal.
Left \(R[G]\)-modules are exactly \(R\)-modules equipped with an \(R\)-linear action of \(G\). In particular, if \(k\) is a field, linear representations \[ \rho:G\to GL(V) \] are equivalent to left \(k[G]\)-module structures on \(V\).
When \(G\) is finite and \(\operatorname{char}k\nmid |G|\), Maschke’s theorem says that \(k[G]\) is semisimple. Over an algebraically closed field of characteristic not dividing \(|G|\), Artin–Wedderburn therefore decomposes \(k[G]\) as a product of matrix algebras indexed by irreducible representations.