Theorems · Theorem · commutative algebra

Valuation.Integers.isPrincipal_iff_exists_eq_setOf_valuation_le

Deprecated since 2026-07-09Use Valuation.Integers.isPrincipal_iff_exists_eq_setOfPred_valuation_le instead.

∀ {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, ↑I = {y | v ((algebraMap O F) y) ≤ v ((algebraMap O F) x)}

Alias of Valuation.Integers.isPrincipal_iff_exists_eq_setOfPred_valuation_le.

Defined in

Mathlib.RingTheory.Valuation.Integers

Cited by

0 results in Mathlib

Foundations

Depth 49 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

FieldLinearOrderedCommGroupWithZeroCommRingAlgebra