Theorems · Theorem · ring theory
IsSimpleModule.nontrivial
∀ (R : Type u_2) [inst : Ring R] (M : Type u_4) [inst_1 : AddCommGroup M] [inst_2 : Module R M] [IsSimpleModule R M], Nontrivial M
- Defined in
- Mathlib.RingTheory.SimpleModule.Basic
- Cited by
- 8 results in Mathlib
- Foundations
- Depth 65 from the axioms · uses propext, Classical.choice, Quot.sound
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
- AddCommGroupstatement and proof · cited by 12,871
- Top.topproof · cited by 9,680
- Ringstatement and proof · cited by 7,463
- Bot.botproof · cited by 4,720
- Nontrivialstatement · cited by 2,416
- Submodule.extproof · cited by 204
- IsSimpleModulestatement and proof · cited by 114
- bot_ne_topproof · cited by 24
Cited by8
Results whose statement or proof uses this declaration.
- isSimpleModule_iff_quot_maximalproof · cited by 5
- bot_lt_isotypicComponentproof · cited by 3
- Submodule.le_linearEquiv_of_sSup_eq_topproof · cited by 2
- isSimpleModule_iff_toSpanSingleton_surjectiveproof · cited by 1
- IsIsotypicOfType.of_subsingletonproof · cited by 1
- IsSemisimpleModule.exists_linearEquiv_fin_dfinsuppproof · cited by 1
- IsSimpleModule.finrank_eq_one_of_isMulCommutativeproof · cited by 1
- isSimpleModule_iff_finrank_eq_oneproof · cited by 0