Theorems · Theorem · commutative algebra
Ideal.IsPrime.mul_le
∀ {R : Type u} [inst : CommSemiring R] {I J P : Ideal R}, P.IsPrime → (I * J ≤ P ↔ I ≤ P ∨ J ≤ P)- Defined in
- Mathlib.RingTheory.Ideal.Operations
- Cited by
- 8 results in Mathlib
- Foundations
- Depth 71 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.
Cites5
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.mul_leproof · cited by 11
- Ideal.IsPrime.mul_mem_iff_mem_or_memproof · cited by 8
Cited by8
Results whose statement or proof uses this declaration.
- Ideal.prime_of_isPrimeproof · cited by 15
- Ideal.IsPrime.multiset_prod_leproof · cited by 3
- Ideal.IsPrime.inf_leproof · cited by 3
- PrimeSpectrum.zeroLocus_mulproof · cited by 1
- Polynomial.mul_contentIdeal_le_radical_contentIdeal_mulproof · cited by 1
- Submodule.exists_eq_colon_of_mem_minimalPrimesproof · cited by 1
- ProjectiveSpectrum.zeroLocus_mul_homogeneousIdealproof · cited by 0
- ProjectiveSpectrum.zeroLocus_mul_idealproof · cited by 0