Theorems · Theorem · ring theory
isSimpleModule_iff_toSpanSingleton_surjective
∀ {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 ∧ ∀ (x : M), x ≠ 0 → Function.Surjective ⇑(LinearMap.toSpanSingleton R M x)- Defined in
- Mathlib.RingTheory.SimpleModule.Basic
- Cited by
- 1 results in Mathlib
- Foundations
- Depth 68 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.
Cites19
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- DFunLike.coestatement and proof · cited by 62,936
- Modulestatement and proof · cited by 20,661
- RingHom.idstatement · cited by 18,349
- AddCommGroupstatement and proof · cited by 12,871
- LinearMapstatement · cited by 10,215
- Top.topproof · cited by 9,680
- Ringstatement and proof · cited by 7,463
- Submoduleproof · cited by 7,192
- Bot.botproof · cited by 4,720
- Nontrivialstatement and proof · cited by 2,416
- LinearMap.idproof · cited by 625
- Submodule.smul_memproof · cited by 204
Cited by1
Results whose statement or proof uses this declaration.
- isSimpleModule_self_iff_isUnitproof · cited by 0