Theorems · Theorem · commutative algebra
IsLocalization.AtPrime.isLocalRing
∀ {R : Type u_1} [inst : CommSemiring R] (S : Type u_2) [inst_1 : CommSemiring S] [inst_2 : Algebra R S] (P : Ideal R)
[hp : P.IsPrime] [IsLocalization.AtPrime S P], IsLocalRing S- Cited by
- 19 results in Mathlib
- Foundations
- Depth 33 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites26
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Algebrastatement and proof · cited by 11,388
- CommSemiringstatement and proof · cited by 10,911
- Idealstatement and proof · cited by 4,748
- mul_oneproof · cited by 3,885
- one_mulproof · cited by 2,841
- IsUnitproof · cited by 1,602
- Ideal.IsPrimestatement and proof · cited by 827
- Ideal.primeComplproof · cited by 462
- IsLocalRingstatement · cited by 339
- IsLocalization.mk'proof · cited by 218
- Ideal.mul_mem_leftproof · cited by 107
- IsLocalization.AtPrimestatement and proof · cited by 79
Cited by19
Results whose statement or proof uses this declaration.
- Localization.AtPrime.under_maximalIdealproof · cited by 9
- IsLocalization.AtPrime.to_map_mem_maximal_iffstatement · cited by 9
- IsLocalization.AtPrime.map_eq_maximalIdealproof · cited by 8
- IsLocalization.AtPrime.under_maximalIdealstatement and proof · cited by 7
- Ideal.algebraMap_residueField_eq_zeroproof · cited by 5
- IsLocalization.AtPrime.ringKrullDim_eq_heightproof · cited by 4
- IsLocalization.AtPrime.isUnit_mk'_iffproof · cited by 3
- IsLocalization.AtPrime.isDiscreteValuationRing_of_dedekind_domainproof · cited by 2
- Algebra.QuasiFiniteAt.exists_basicOpen_eq_singletonproof · cited by 2
- IsLocalization.AtPrime.not_isFieldproof · cited by 2
- IsArtinianRing.exists_not_mem_forall_mem_of_neproof · cited by 1
- AlgHom.IsArithFrobAt.isArithFrobAt_localizeproof · cited by 1