Theorems · Definition · ring theory

Module.End.IsFinitelySemisimple

{R : Type u_1} →
  {M : Type u_2} → [inst : CommRing R] → [inst_1 : AddCommGroup M] → [inst_2 : Module R M] → Module.End R M → Prop

A weaker version of semisimplicity that only prescribes behaviour on finitely-generated submodules.

Defined in: Mathlib.LinearAlgebra.Semisimple
Cited by: 12 results in Mathlib
Foundations: Depth 112 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes: CommRing AddCommGroup Module

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Around this declaration

Dashed lines are statement dependencies; solid lines are citations in proofs.

Cites9

Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.

Modulestatement and proof · cited by 20,661
CommRingstatement and proof · cited by 17,173
AddCommGroupstatement and proof · cited by 12,871
Submoduleproof · cited by 7,192
Module.Finiteproof · cited by 1,032
Module.Endstatement and proof · cited by 774
Module.End.invtSubmoduleproof · cited by 93
LinearMap.restrictproof · cited by 84
Module.End.IsSemisimpleproof · cited by 34

Cited by12

Results whose statement or proof uses this declaration.

Module.End.IsFinitelySemisimple.maxGenEigenspace_eq_eigenspacestatement and proof · cited by 4
Module.End.IsSemisimple.isFinitelySemisimplestatement · cited by 4
Module.End.isFinitelySemisimple_iffstatement · cited by 2
Module.End.apply_eq_of_mem_of_comm_of_isFinitelySemisimple_of_isNilstatement and proof · cited by 2
Module.End.isFinitelySemisimple_iff'statement · cited by 1
Module.End.isFinitelySemisimple_iff_isSemisimplestatement and proof · cited by 1
Module.End.IsFinitelySemisimple.genEigenspace_eq_eigenspacestatement and proof · cited by 1
Module.End.IsFinitelySemisimple.iSup_maxGenEigenspace_eq_top_iffstatement and proof · cited by 1
Module.End.IsFinitelySemisimple.restrictstatement and proof · cited by 1
Module.End.eq_zero_of_isNilpotent_of_isFinitelySemisimplestatement and proof · cited by 1
LinearMap.IsSymmetric.isFinitelySemisimplestatement · cited by 1
Module.End.isFinitelySemisimple_sub_algebraMap_iffstatement · cited by 0