Theorems · Theorem · commutative algebra
Ideal.le_of_dvd
∀ {R : Type u} [inst : CommSemiring R] {I J : Ideal R}, I ∣ J → J ≤ IIf I divides J, then I contains J.
In a Dedekind domain, to divide and contain are equivalent, see Ideal.dvd_iff_le.
- Defined in
- Mathlib.RingTheory.Ideal.Operations
- Cited by
- 9 results in Mathlib
- Foundations
- Depth 75 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
- le_transproof · cited by 985
- inf_le_leftproof · cited by 286
- Ideal.mul_le_infproof · cited by 10
Cited by9
Results whose statement or proof uses this declaration.
- Ideal.dvd_iff_leproof · cited by 33
- Ideal.prime_of_isPrimeproof · cited by 15
- Ideal.isUnit_iffproof · cited by 7
- Ideal.IsDedekindDomain.ramificationIdx'_eq_normalizedFactors_countproof · cited by 5
- Ideal.count_normalizedFactors_eqproof · cited by 3
- Ideal.ramificationIdx'_algebra_towerproof · cited by 2
- IsDedekindDomain.HeightOneSpectrum.mem_integers_of_valuation_le_oneproof · cited by 1
- RingOfIntegers.isPrincipalIdealRing_of_isPrincipal_of_norm_le_of_isPrimeproof · cited by 1
- Ideal.Factors.ramificationIdx_ne_zeroproof · cited by 0