Theorems · Inductive type · ring theory
IsSemisimpleModule
(R : Type u_2) → [inst : Ring R] → (M : Type u_4) → [inst_1 : AddCommGroup M] → [Module R M] → Prop
A module is semisimple when every submodule has a complement, or equivalently, the module is a direct sum of simple modules.
- Defined in
- Mathlib.RingTheory.SimpleModule.Basic
- Cited by
- 67 results in Mathlib
- Foundations
- Depth 6 from the axioms · uses no axioms
- Assumes
- RingAddCommGroupModule
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites3
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Modulestatement · cited by 20,661
- AddCommGroupstatement · cited by 12,871
- Ringstatement · cited by 7,463
Cited by74
Results whose statement or proof uses this declaration.
- Module.End.IsSemisimpleproof · cited by 34
- IsSemisimpleRingproof · cited by 28
- IsSemisimpleModule.congrstatement and proof · cited by 6
- isSemisimpleModule_iffstatement and proof · cited by 5
- IsSemisimpleModule.annihilator_isRadicalstatement and proof · cited by 4
- IsSemisimpleModule.eq_bot_or_exists_simple_lestatement and proof · cited by 4
- IsSemisimpleModule.jacobson_eq_botstatement and proof · cited by 4
- IsSemisimpleModule.sSup_simples_eq_topstatement and proof · cited by 4
- LinearMap.isSemisimpleModule_iff_of_bijectivestatement · cited by 4
- IsSemisimpleModule.exists_end_algEquiv_pi_matrix_endstatement and proof · cited by 3
- IsSemisimpleModule.exists_linearEquiv_dfinsuppstatement and proof · cited by 3
- OrderIso.setIsotypicComponentsstatement and proof · cited by 3