Theorems · Theorem · commutative algebra
IsLocalRing.eq_maximalIdeal
∀ {R : Type u_1} [inst : CommSemiring R] [inst_1 : IsLocalRing R] {I : Ideal R},
I.IsMaximal → I = IsLocalRing.maximalIdeal R- Cited by
- 20 results in Mathlib
- Foundations
- Depth 34 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- CommSemiringIsLocalRing
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites7
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- CommSemiringstatement and proof · cited by 10,911
- Idealstatement and proof · cited by 4,748
- Ideal.IsMaximalstatement and proof · cited by 452
- IsLocalRingstatement and proof · cited by 339
- IsLocalRing.maximalIdealstatement · cited by 297
- ExistsUnique.uniqueproof · cited by 42
- IsLocalRing.maximal_ideal_uniqueproof · cited by 2
Cited by20
Results whose statement or proof uses this declaration.
- IsLocalRing.le_maximalIdealproof · cited by 20
- IsLocalRing.maximalIdeal_le_jacobsonproof · cited by 16
- Algebra.FormallyUnramified.map_maximalIdealproof · cited by 5
- IsDiscreteValuationRing.irreducible_iff_uniformizerproof · cited by 4
- AdicCompletion.maximalIdeal_eq_map_of_fgproof · cited by 3
- Module.FaithfullyFlat.of_flat_of_isLocalHomproof · cited by 3
- AdicCompletion.maximalIdeal_eq_mapproof · cited by 2
- Ideal.exists_ideal_over_prime_of_isIntegral_of_isDomainproof · cited by 2
- CovBy.length_baseChangeproof · cited by 1
- CovBy.length_restrictScalarsproof · cited by 1
- maximalIdeal_isPrincipal_of_isDedekindDomainproof · cited by 1
- IsLocalRing.not_isLocalRing_tfaeproof · cited by 1