Theorems · Theorem · commutative algebra
RingHom.injective_iff_ker_eq_bot
∀ {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), Function.Injective ⇑f ↔ RingHom.ker f = ⊥- Defined in
- Mathlib.RingTheory.Ideal.Maps
- Cited by
- 29 results in Mathlib
- Foundations
- Depth 20 from the axioms · uses propext, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites14
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
- Setproof · cited by 53,352
- Semiringstatement and proof · cited by 13,802
- SetLike.coeproof · cited by 8,199
- 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 · cited by 363
- RingHomClassstatement and proof · cited by 193
- Set.ext_iffproof · cited by 90
- SetLike.ext'_iffproof · cited by 78
Cited by29
Results whose statement or proof uses this declaration.
- RingHom.ker_eq_bot_iff_eq_zeroproof · cited by 7
- Ideal.sum_ramification_inertiaproof · cited by 4
- Ideal.exists_maximal_ideal_liesOver_of_isIntegralproof · cited by 3
- Algebra.trace_quotient_eq_of_isDedekindDomainproof · cited by 3
- RingHom.ker_comp_of_injectiveproof · cited by 3
- RingHom.lift_injective_of_ker_le_idealproof · cited by 2
- Ideal.exists_ideal_over_prime_of_isIntegral_of_isPrimeproof · cited by 2
- AlgebraicGeometry.Scheme.Hom.app_injectiveproof · cited by 1
- AlgebraicGeometry.injective_germ_basicOpenproof · cited by 1
- MvPolynomial.eq_zero_of_eval_zero_at_prod_finsetproof · cited by 1
- Ideal.injective_algebraMap_quotient_residueFieldproof · cited by 1
- Ideal.injective_lift_iffproof · cited by 1