Theorems · Theorem · commutative algebra
RingHom.ker_eq_bot_iff_eq_zero
∀ {R : Type u} {S : Type v} {F : Type u_1} [inst : Ring R] [inst_1 : Semiring S] [inst_2 : FunLike F R S]
[rc : RingHomClass F R S] (f : F), RingHom.ker f = ⊥ ↔ ∀ (x : R), f x = 0 → x = 0- Defined in
- Mathlib.RingTheory.Ideal.Maps
- Cited by
- 7 results in Mathlib
- Foundations
- Depth 21 from the axioms · uses propext, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites10
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- DFunLike.coestatement · cited by 62,936
- Semiringstatement and proof · cited by 13,802
- Ringstatement and proof · cited by 7,463
- Idealstatement · cited by 4,748
- Bot.botstatement and proof · cited by 4,720
- FunLikestatement and proof · cited by 2,560
- RingHom.kerstatement and proof · cited by 363
- RingHomClassstatement and proof · cited by 193
- injective_iff_map_eq_zeroproof · cited by 62
- RingHom.injective_iff_ker_eq_botproof · cited by 29
Cited by7
Results whose statement or proof uses this declaration.
- Algebra.trace_quotient_eq_of_isDedekindDomainproof · cited by 3
- RingHom.lift_injective_of_ker_le_idealproof · cited by 2
- NumberField.RingOfIntegers.ker_algebraMap_eq_botproof · cited by 1
- AlgebraicGeometry.injective_germ_basicOpenproof · cited by 1
- AlgebraicGeometry.stalkMap_injective_of_isOpenMap_of_injectiveproof · cited by 0
- AlgebraicGeometry.Scheme.IsGermInjective.Specproof · cited by 0
- Module.IsTorsionFree.of_isLocalizationproof · cited by 0