Theorems · Theorem · commutative algebra
Valuation.Integers.isPrincipal_iff_exists_isGreatest
∀ {F : Type u} {Γ₀ : Type v} [inst : Field F] [inst_1 : LinearOrderedCommGroupWithZero Γ₀] {v : Valuation F Γ₀}
{O : Type w} [inst_2 : CommRing O] [inst_3 : Algebra O F],
v.Integers O → ∀ {I : Ideal O}, Submodule.IsPrincipal I ↔ ∃ x, IsGreatest (⇑v ∘ ⇑(algebraMap O F) '' ↑I) x- Defined in
- Mathlib.RingTheory.Valuation.Integers
- Cited by
- 2 results in Mathlib
- Foundations
- Depth 47 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites24
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- DFunLike.coestatement and proof · cited by 62,936
- CommRingstatement and proof · cited by 17,173
- Algebrastatement and proof · cited by 11,388
- RingHomstatement · cited by 10,189
- SetLike.coestatement and proof · cited by 8,199
- Fieldstatement and proof · cited by 7,404
- Set.imagestatement and proof · cited by 5,609
- Idealstatement and proof · cited by 4,748
- Algebra.algebraMapstatement and proof · cited by 4,706
- Submodule.spanproof · cited by 1,504
- Ideal.spanproof · cited by 948
- Valuationstatement and proof · cited by 823
Cited by2
Results whose statement or proof uses this declaration.
- Valuation.Integers.isPrincipal_iff_exists_eq_setOfPred_valuation_leproof · cited by 1
- Valuation.Integers.not_denselyOrdered_of_isPrincipalIdealRingproof · cited by 1