Theorems · Definition · commutative algebra
Ideal.radical
{R : Type u} → [inst : CommSemiring R] → Ideal R → Ideal RThe radical of an ideal I consists of the elements r such that r ^ n ∈ I for some n.
- Defined in
- Mathlib.RingTheory.Ideal.Operations
- Cited by
- 121 results in Mathlib
- Foundations
- Depth 76 from the axioms, rests on 1,774 definitions · uses propext, Classical.choice, Quot.sound
- Assumes
- CommSemiring
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites3
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
- Set.ofPredproof · cited by 6,101
- Idealstatement and proof · cited by 4,748
Cited by137
Results whose statement or proof uses this declaration.
- nilradicalproof · cited by 41
- Ideal.IsRadicalproof · cited by 30
- Ideal.radical_eq_sInfstatement and proof · cited by 21
- Ideal.le_radicalstatement · cited by 16
- Ideal.IsPrime.isRadicalproof · cited by 14
- AlgebraicGeometry.Scheme.IdealSheafData.radicalproof · cited by 10
- Ideal.radical_monostatement and proof · cited by 10
- Ideal.IsPrime.radical_le_iffstatement · cited by 9
- Submodule.IsAssociatedPrime.casesOnstatement and proof · cited by 8
- Ideal.IsRadical.radicalstatement · cited by 8
- Ideal.radical_isRadicalstatement and proof · cited by 7
- PrimeSpectrum.vanishingIdeal_zeroLocus_eq_radicalstatement and proof · cited by 7