Theorems · Theorem · ring theory
isSimpleModule_iff
∀ (R : Type u_2) [inst : Ring R] (M : Type u_4) [inst_1 : AddCommGroup M] [inst_2 : Module R M], IsSimpleModule R M ↔ IsSimpleOrder (Submodule R M)
- Defined in
- Mathlib.RingTheory.SimpleModule.Basic
- Cited by
- 12 results in Mathlib
- Foundations
- Depth 67 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- RingAddCommGroupModule
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites7
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
- AddCommGroupstatement and proof · cited by 12,871
- Ringstatement and proof · cited by 7,463
- Submodulestatement and proof · cited by 7,192
- IsSimpleModulestatement and proof · cited by 114
- IsSimpleOrderstatement and proof · cited by 54
- IsSimpleModule.casesOnproof · cited by 1
Cited by12
Results whose statement or proof uses this declaration.
- isSimpleModule_iff_isAtomproof · cited by 7
- isSimpleModule_iff_isCoatomproof · cited by 6
- Module.length_eq_one_iffproof · cited by 4
- isSimpleModule_iff_toSpanSingleton_surjectiveproof · cited by 1
- RootPairing.isSimpleModule_weylGroupRootRep_iffproof · cited by 1
- Representation.isSimpleModule_iff_irreducible_ofModuleproof · cited by 1
- simple_iff_isSimpleModuleproof · cited by 1
- isSimpleModule_iff_isSimpleModule_of_algebraMap_surjectiveproof · cited by 1
- Representation.irreducible_iff_isSimpleModule_asModuleproof · cited by 0
- simple_of_finrank_eq_oneproof · cited by 0
- isSimpleModule_iff_eq_zero_or_injectiveproof · cited by 0
- isSimpleModule_iff_finrank_eq_oneproof · cited by 0