Theorems · Theorem · commutative algebra
Ideal.algebraMap_residueField_eq_zero
∀ {R : Type u_1} [inst : CommRing R] {I : Ideal R} [inst_1 : I.IsPrime] {x : R},
(algebraMap R I.ResidueField) x = 0 ↔ x ∈ I- Cited by
- 5 results in Mathlib
- Foundations
- Depth 99 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- CommRingIdeal.IsPrime
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites15
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
- CommRingstatement and proof · cited by 17,173
- RingHomstatement and proof · cited by 10,189
- Idealstatement and proof · cited by 4,748
- Algebra.algebraMapstatement and proof · cited by 4,706
- Ideal.IsPrimestatement and proof · cited by 827
- Ideal.primeComplstatement · cited by 462
- Localization.AtPrimestatement and proof · cited by 299
- IsLocalRing.ResidueFieldstatement and proof · cited by 156
- Ideal.ResidueFieldstatement and proof · cited by 119
- IsScalarTower.algebraMap_applyproof · cited by 116
- IsLocalization.AtPrime.isLocalRingproof · cited by 19
Cited by5
Results whose statement or proof uses this declaration.
- Ideal.ker_algebraMap_residueFieldproof · cited by 5
- Algebra.IsUnramifiedAt.exists_notMem_forall_ne_mem_and_adjoin_eq_topproof · cited by 1
- Algebra.exists_etale_isIdempotentElem_forall_liesOver_eq_auxproof · cited by 1
- PrimeSpectrum.residueField_comapproof · cited by 0
- Algebra.IsUnramifiedAt.not_minpoly_sq_dvdproof · cited by 0