Editing Group ring (section)

===Semisimple decomposition===
The dimension of the vector space ''K''[''G''] is just equal to the number of elements in the group. The field ''K'' is commonly taken to be the complex numbers '''C''' or the reals '''R''', so that one discusses the group algebras '''C'''[''G''] or '''R'''[''G''].

The group algebra '''C'''[''G''] of a finite group over the complex numbers is a [[semisimple ring]]. This result, [[Maschke's theorem]], allows us to understand '''C'''[''G''] as a finite [[Product of rings|product]] of [[matrix ring]]s with entries in '''C'''. Indeed, if we list the complex [[irreducible representation]]s of ''G'' as ''V<sub>k</sub>'' for ''k'' = 1, . . . , ''m'', these correspond to [[group homomorphism]]s <math>\rho_k: G\to \mathrm{Aut}(V_k)</math> and hence to algebra homomorphisms <math>\tilde\rho_k: \mathbb{C}[G]\to \mathrm{End}(V_k)</math>. Assembling these mappings gives an algebra isomorphism
:<math>\tilde\rho : \mathbb{C}[G] \to \bigoplus_{k=1}^m \mathrm{End}(V_k)
\cong \bigoplus_{k=1}^m M_{d_k}(\mathbb{C}),  </math>
where ''d<sub>k</sub>'' is the dimension of ''V<sub>k</sub>''. The subalgebra of '''C'''[''G''] corresponding to End(''V<sub>k</sub>'') is the [[Ideal (ring theory)|two-sided ideal]] generated by the [[Idempotent (ring theory)|idempotent]]
:<math>\epsilon_k = \frac{d_k}{|G|}\sum_{g\in G}\chi_k(g^{-1})\,g,   </math>
where <math>\chi_k(g)=\mathrm{tr}\,\rho_k(g)  </math> is the [[Character theory|character]] of ''V<sub>k</sub>''. These form a complete system of orthogonal idempotents, so that <math>\epsilon_k^2 =\epsilon_k  </math>, <math>\epsilon_j \epsilon_k = 0  </math> for ''j ≠ k'', and <math>1 = \epsilon_1+\cdots+\epsilon_m   </math>. The isomorphism <math>\tilde\rho</math> is closely related to [[Fourier transform on finite groups]].

For a more general field ''K,'' whenever the [[characteristic (algebra)|characteristic]] of ''K'' does not divide the order of the group ''G'', then ''K''[''G''] is semisimple. When ''G'' is a finite [[abelian group]], the group ring ''K''[G] is commutative, and its structure is easy to express in terms of [[root of unity|roots of unity]].

When ''K'' is a field of characteristic ''p'' which divides the order of ''G'', 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.