Theorems · Theorem · commutative algebra

Valuation.HasExtension.ofComapInteger

∀ {A : Type u_2} [inst : Ring A] {K : Type u_5} [inst_1 : Field K] [inst_2 : Algebra K A] {ΓA : Type u_7}
  {ΓK : Type u_8} [inst_3 : LinearOrderedCommGroupWithZero ΓK] [inst_4 : LinearOrderedCommGroupWithZero ΓA]
  {vK : Valuation K ΓK} {vA : Valuation A ΓA},
  Subring.comap (algebraMap K A) vA.integer = vK.integer → vK.HasExtension vA

When K is a field, if the preimage of the valuation integers of A equals to the valuation integers of K, then the valuation on A is an extension of the valuation on K.

Defined in

Mathlib.RingTheory.Valuation.Extension

Cited by

0 results in Mathlib

Foundations

Depth 48 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

RingFieldAlgebraLinearOrderedCommGroupWithZeroLinearOrderedCommGroupWithZero