Theorems · Theorem · commutative algebra
Ideal.IsPrime.radical_le_iff
∀ {R : Type u} [inst : CommSemiring R] {I J : Ideal R}, J.IsPrime → (I.radical ≤ J ↔ I ≤ J)- Defined in
- Mathlib.RingTheory.Ideal.Operations
- Cited by
- 9 results in Mathlib
- Foundations
- Depth 82 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- CommSemiring
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.
- CommSemiringstatement and proof · cited by 10,911
- Idealstatement and proof · cited by 4,748
- Ideal.IsPrimestatement and proof · cited by 827
- Ideal.radicalstatement · cited by 121
- Ideal.IsPrime.isRadicalproof · cited by 14
- Ideal.IsRadical.radical_le_iffproof · cited by 5
Cited by9
Results whose statement or proof uses this declaration.
- Ideal.radical_eq_sInfproof · cited by 21
- Ideal.radical_eq_jacobsonproof · cited by 2
- Ideal.iUnion_minimalPrimesproof · cited by 1
- Submodule.IsMinimalPrimaryDecomposition.comap_localized₀_eq_iteproof · cited by 1
- Ideal.radical_minimalPrimesproof · cited by 1
- PrimeSpectrum.denseRange_comap_iff_minimalPrimesproof · cited by 0
- Ideal.isPrimary_of_isMaximal_radicalproof · cited by 0
- Ideal.isLocal_of_isMaximal_radicalproof · cited by 0
- Ideal.minimalPrimes_eq_subsingletonproof · cited by 0