Theorems · Theorem · commutative algebra
Ideal.IsRadical.radical_le_iff
∀ {R : Type u} [inst : CommSemiring R] {I J : Ideal R}, J.IsRadical → (I.radical ≤ J ↔ I ≤ J)- Defined in
- Mathlib.RingTheory.Ideal.Operations
- Cited by
- 5 results in Mathlib
- Foundations
- Depth 81 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.
Cites8
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
- le_transproof · cited by 985
- Ideal.radicalstatement · cited by 121
- Ideal.IsRadicalstatement and proof · cited by 30
- Ideal.le_radicalproof · cited by 16
- Ideal.radical_monoproof · cited by 10
- Ideal.IsRadical.radicalproof · cited by 8
Cited by5
Results whose statement or proof uses this declaration.
- Ideal.IsPrime.radical_le_iffproof · cited by 9
- Ideal.radical_le_radical_iffproof · cited by 4
- PrimeSpectrum.isIrreducible_zeroLocus_iff_of_radicalproof · cited by 3
- Ideal.mem_minimalPrimes_span_of_mem_minimalPrimes_span_insertproof · cited by 1
- ringKrullDim_eq_one_iff_of_isLocalRing_isDomainproof · cited by 0