Theorems · Theorem · commutative algebra
CharacterModule.dual_surjective_iff_injective
∀ {R : Type uR} [inst : CommRing R] {A : Type uA} [inst_1 : AddCommGroup A] {A' : Type u_1} [inst_2 : AddCommGroup A']
[inst_3 : Module R A] [inst_4 : Module R A'] {f : A →ₗ[R] A'},
Function.Surjective ⇑(CharacterModule.dual f) ↔ Function.Injective ⇑f- Defined in
- Mathlib.Algebra.Module.CharacterModule
- Cited by
- 1 results in Mathlib
- Foundations
- Depth 98 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites12
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 and proof · cited by 18,349
- CommRingstatement and proof · cited by 17,173
- AddCommGroupstatement and proof · cited by 12,871
- LinearMapstatement and proof · cited by 10,215
- injective_iff_map_eq_zeroproof · cited by 62
- AddMonoidHom.map_zeroproof · cited by 47
- CharacterModulestatement and proof · cited by 26
- CharacterModule.dualstatement and proof · cited by 13
- CharacterModule.dual_surjective_of_injectiveproof · cited by 2
- CharacterModule.eq_zero_of_character_applyproof · cited by 2
Cited by1
Results whose statement or proof uses this declaration.
- CharacterModule.dual_bijective_iff_bijectiveproof · cited by 1