Section IrrComponent.
(* The component of the socle of the regular module that is associated to an *)
(* irreducible representation.                                               *)
Variables (n : nat) (rG : mx_representation F G n).
Local Notation E_G := (enveloping_algebra_mx rG).
Let not_rsim_op0 (iG j : sG) A :
mx_rsim rG (socle_repr iG) -> iG != j -> (A \in 'R_j)%MS ->
gring_op rG A = 0.
Definition irr_comp := odflt 1%irr [pick i | gring_op rG 'e_i != 0].
Local Notation iG := irr_comp.
Hypothesis irrG : mx_irreducible rG.
End IrrComponent.

Lemma irr_comp_id (M : 'M_nG) (modM : mxmodule aG M) (iM : sG) :
mxsimple aG M -> (M <= iM)%MS -> irr_comp (submod_repr modM) = iM.