Local Notation "''R_' i" := (Wedderburn_subring i) : group_ring_scope.
Let sums_R : (\sum_i 'R_i :=: Socle sG *m R_G)%MS.

(* Specifications: *)
Lemma Wedderburn_ideal i : mx_ideal R_G 'R_i.
Lemma Wedderburn_direct : mxdirect (\sum_i 'R_i)%MS.
Lemma Wedderburn_disjoint i j : i != j -> ('R_i :&: 'R_j)%MS = 0.
Lemma Wedderburn_annihilate i j : i != j -> ('R_i * 'R_j)%MS = 0.
Lemma Wedderburn_mulmx0 i j A B :
i != j -> (A \in 'R_i)%MS -> (B \in 'R_j)%MS -> A *m B = 0.