Theorems · Theorem · commutative algebra
Ideal.IsPrime.isRadical
∀ {R : Type u} [inst : CommSemiring R] {I : Ideal R}, I.IsPrime → I.IsRadical- Defined in
- Mathlib.RingTheory.Ideal.Operations
- Cited by
- 14 results in Mathlib
- Foundations
- Depth 78 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.radicalproof · cited by 121
- Ideal.IsRadicalstatement · cited by 30
- Ideal.IsPrime.mem_of_pow_memproof · cited by 10
Cited by14
Results whose statement or proof uses this declaration.
- Ideal.disjoint_powers_iff_notMem_of_isPrimeproof · cited by 10
- Ideal.IsPrime.radical_le_iffproof · cited by 9
- isJacobsonRing_iff_prime_eqproof · cited by 5
- Ideal.IsPrime.radicalproof · cited by 5
- IsSemisimpleModule.annihilator_isRadicalproof · cited by 4
- isJacobsonRing_of_surjectiveproof · cited by 4
- IsLocalization.isMaximal_iff_isMaximal_disjointproof · cited by 3
- isJacobsonRing_iff_sInf_maximalproof · cited by 1
- isJacobsonRing_localizationproof · cited by 1
- PrimeSpectrum.vanishingIdeal_isIrreducibleproof · cited by 1
- Ideal.mem_minimalPrimes_span_of_mem_minimalPrimes_span_insertproof · cited by 1
- Algebra.exists_etale_isIdempotentElem_forall_liesOver_eqproof · cited by 1