Theorems · Theorem · commutative algebra
IsLocalRing.jacobson_eq_maximalIdeal
∀ {R : Type u_1} [inst : CommRing R] [inst_1 : IsLocalRing R] (I : Ideal R),
I ≠ ⊤ → I.jacobson = IsLocalRing.maximalIdeal R- Cited by
- 10 results in Mathlib
- Foundations
- Depth 76 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- CommRingIsLocalRing
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.
- CommRingstatement and proof · cited by 17,173
- Top.topstatement and proof · cited by 9,680
- Idealstatement and proof · cited by 4,748
- le_antisymmproof · cited by 2,068
- IsLocalRingstatement and proof · cited by 339
- IsLocalRing.maximalIdealstatement · cited by 297
- sInf_leproof · cited by 110
- Ideal.jacobsonstatement · cited by 88
- IsLocalRing.le_maximalIdealproof · cited by 20
- IsLocalRing.maximalIdeal_le_jacobsonproof · cited by 16
Cited by10
Results whose statement or proof uses this declaration.
- IsLocalRing.ringJacobson_eq_maximalIdealproof · cited by 4
- IsLocalRing.exists_maximalIdeal_pow_le_of_isArtinianRing_quotientproof · cited by 2
- Algebra.FormallyUnramified.bijective_of_isAlgClosed_of_isLocalRingproof · cited by 1
- IsLocalRing.quotient_span_eq_top_iff_span_eq_topproof · cited by 1
- IsLocalRing.rank_cotangentSpace_eq_spanrank_maximalIdeal_of_fgproof · cited by 1
- Ideal.height_le_one_of_isPrincipal_of_mem_minimalPrimes_of_isLocalRingproof · cited by 1
- IsLocalRing.isRegular_iff_isWeaklyRegular_of_subset_maximalIdealproof · cited by 1
- Algebra.FormallyUnramified.isField_quotient_map_maximalIdealproof · cited by 1
- IsLocalRing.CotangentSpace.map_eq_top_iffproof · cited by 1
- IsLocalRing.map_mkQ_eqproof · cited by 1