Theorems · Theorem · commutative algebra
Ideal.dvd_iff_le
∀ {A : Type u_2} [inst : CommRing A] [IsDedekindDomain A] {I J : Ideal A}, I ∣ J ↔ J ≤ IFor ideals in a Dedekind domain, to divide is to contain.
- Cited by
- 33 results in Mathlib
- Foundations
- Depth 144 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- CommRingIsDedekindDomain
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites21
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- CommRingstatement and proof · cited by 17,173
- Idealstatement and proof · cited by 4,748
- Bot.botproof · cited by 4,720
- one_mulproof · cited by 2,841
- le_reflproof · cited by 2,061
- mul_assocproof · cited by 1,667
- nonZeroDivisorsproof · cited by 895
- IsDedekindDomainstatement and proof · cited by 668
- FractionalIdealproof · cited by 423
- mul_le_mul'proof · cited by 274
- inv_mul_cancel₀proof · cited by 267
- mul_inv_cancel₀proof · cited by 210
Cited by33
Results whose statement or proof uses this declaration.
- Ideal.span_singleton_dvd_span_singleton_iff_dvdproof · cited by 6
- Ideal.dvd_span_singletonproof · cited by 5
- Ideal.absNorm_dvd_absNorm_of_leproof · cited by 5
- Ideal.IsDedekindDomain.ramificationIdx'_eq_normalizedFactors_countproof · cited by 5
- RingOfIntegers.not_dvd_exponent_iffproof · cited by 4
- Ideal.count_associates_factors_eqproof · cited by 4
- IsDedekindDomain.idealFactorsEquivOfQuotEquiv_is_dvd_isoproof · cited by 3
- Ideal.mul_iInfproof · cited by 3
- Ideal.sup_eq_prod_inf_factorsproof · cited by 3
- Ideal.mem_normalizedFactors_iffproof · cited by 3
- Ideal.IsDedekindDomain.ramificationIdx'_eq_one_iffproof · cited by 3
- Ideal.count_le_of_ideal_geproof · cited by 3