Theorems · Theorem · commutative algebra
Ideal.isCoprime_iff_gcd
∀ {A : Type u_2} [inst : CommRing A] [inst_1 : IsDedekindDomain A] {I J : Ideal A}, IsCoprime I J ↔ gcd I J = 1- Cited by
- 2 results in Mathlib
- Foundations
- Depth 152 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.
Cites10
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
- Top.topproof · cited by 9,680
- Idealstatement and proof · cited by 4,748
- IsDedekindDomainstatement and proof · cited by 668
- IsCoprimestatement · cited by 321
- GCDMonoid.gcdstatement and proof · cited by 143
- Ideal.one_eq_topproof · cited by 83
- codisjoint_iffproof · cited by 53
- Ideal.isCoprime_iff_codisjointproof · cited by 6
- Ideal.gcd_eq_supproof · cited by 1
Cited by2
Results whose statement or proof uses this declaration.
- dvd_differentIdeal_of_not_isSeparableproof · cited by 1
- IsCyclotomicExtension.Rat.isCoprime_of_not_zeta_sub_one_dvdproof · cited by 0