Theorems · Theorem · commutative algebra
Ideal.IsMaximal.eq_of_le
∀ {α : Type u} [inst : Semiring α] {I J : Ideal α}, I.IsMaximal → J ≠ ⊤ → I ≤ J → I = J- Defined in
- Mathlib.RingTheory.Ideal.Maximal
- Cited by
- 39 results in Mathlib
- Foundations
- Depth 26 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- Semiring
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites6
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Semiringstatement and proof · cited by 13,802
- Top.topstatement and proof · cited by 9,680
- Idealstatement and proof · cited by 4,748
- Ideal.IsMaximalstatement and proof · cited by 452
- Ideal.IsMaximal.outproof · cited by 16
- eq_iff_le_not_ltproof · cited by 11
Cited by39
Results whose statement or proof uses this declaration.
- Ring.ne_bot_of_isMaximal_of_not_isFieldproof · cited by 8
- PrimeSpectrum.isClosed_singleton_iff_isMaximalproof · cited by 8
- Ring.krullDimLE_zero_and_isLocalRing_tfaeproof · cited by 7
- Ideal.IsMaximal.coprime_of_neproof · cited by 5
- Valuation.Uniformizer.is_generatorproof · cited by 4
- IsLocalRing.isMaximal_iffproof · cited by 3
- Ring.KrullDimLE.isField_of_isDomainproof · cited by 3
- Ring.DimensionLEOne.not_lt_ltproof · cited by 2
- Ring.isField_iff_maximal_botproof · cited by 2
- Ideal.exists_isMaximal_dvd_of_dvd_absNormproof · cited by 2
- PrimeSpectrum.exists_multiset_prod_cons_le_and_prod_not_leproof · cited by 2
- Ideal.IsMaximal.of_isLocalization_of_disjointproof · cited by 2