Theorems · Theorem · commutative algebra
Ideal.IsPrime.radical
∀ {R : Type u} [inst : CommSemiring R] {I : Ideal R}, I.IsPrime → I.radical = I- 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.
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.radicalproof · cited by 8
Cited by5
Results whose statement or proof uses this declaration.
- Ideal.radical_eq_sInfproof · cited by 21
- Submodule.isAssociatedPrime_iffproof · cited by 1
- PrimeSpectrum.vanishingIdeal_isIrreducibleproof · cited by 1
- Ideal.IsMinimalPrimaryDecomposition.minimalPrimes_subset_image_radicalproof · cited by 0
- MvPolynomial.IsPrime.vanishingIdeal_zeroLocusproof · cited by 0