Theorems · Theorem · commutative algebra
Ideal.span_singleton_eq_span_singleton
∀ {α : Type u} [inst : CommSemiring α] [IsDomain α] {x y : α}, Ideal.span {x} = Ideal.span {y} ↔ Associated x y- Defined in
- Mathlib.RingTheory.Ideal.Span
- Cited by
- 15 results in Mathlib
- Foundations
- Depth 30 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- CommSemiringIsDomain
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites9
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Setstatement · cited by 53,352
- CommSemiringstatement and proof · cited by 10,911
- Idealstatement · cited by 4,748
- IsDomainstatement and proof · cited by 2,196
- Ideal.spanstatement and proof · cited by 948
- Associatedstatement · cited by 296
- le_antisymm_iffproof · cited by 62
- Ideal.span_singleton_le_span_singletonproof · cited by 15
- dvd_dvd_iff_associatedproof · cited by 5
Cited by15
Results whose statement or proof uses this declaration.
- IsDiscreteValuationRing.associated_of_irreducibleproof · cited by 2
- Ideal.torsionOf_eq_span_pow_pOrderproof · cited by 2
- IsCyclotomicExtension.Rat.absNorm_span_zeta_sub_oneproof · cited by 2
- IsDiscreteValuationRing.ideal_eq_span_pow_irreducibleproof · cited by 1
- IsCyclotomicExtension.Rat.map_eq_span_zeta_sub_one_powproof · cited by 1
- NumberField.Units.dirichletUnitTheorem.exists_unitproof · cited by 1
- Nat.span_singleton_setGcdproof · cited by 1
- Int.span_natAbsproof · cited by 1
- IsBezout.span_gcd_eq_span_gcdproof · cited by 1
- IsDiscreteValuationRing.of_ufd_of_unique_irreducibleproof · cited by 1
- Submodule.IsPrincipal.associated_generator_span_selfproof · cited by 0
- Ideal.factors_span_eqproof · cited by 0