Theorems · Theorem · ring theory
isSimpleModule_iff_isAtom
∀ {R : Type u_2} [inst : Ring R] {M : Type u_4} [inst_1 : AddCommGroup M] [inst_2 : Module R M] {m : Submodule R M},
IsSimpleModule R ↥m ↔ IsAtom m- Defined in
- Mathlib.RingTheory.SimpleModule.Basic
- Cited by
- 7 results in Mathlib
- Foundations
- Depth 71 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.
Cites13
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
- Set.Elemproof · cited by 7,166
- Set.Iicproof · cited by 1,111
- IsAtomstatement · cited by 130
- IsSimpleModulestatement and proof · cited by 114
- IsSimpleOrderproof · cited by 54
- isSimpleModule_iffproof · cited by 12
- OrderIso.isSimpleOrder_iffproof · cited by 12
- Submodule.mapIicproof · cited by 10
Cited by7
Results whose statement or proof uses this declaration.
- IsIsotypic.linearEquiv_funproof · cited by 3
- IsSimpleRing.tfaeproof · cited by 1
- Submodule.linearEquiv_of_le_sSupproof · cited by 1
- Submodule.linearEquiv_of_sSup_eq_topproof · cited by 1
- IsIsotypicOfType.of_isSimpleModuleproof · cited by 0
- IsSimpleModule.isAtomproof · cited by 0
- IsIsotypic.linearEquiv_finsuppproof · cited by 0