Theorems · Theorem · commutative algebra

IsFractionRing.ideal_span_singleton_map_subset

∀ (R : Type u_1) [inst : CommRing R] {S : Type u_2} [inst_1 : CommRing S] [inst_2 : Algebra R S] {K : Type u_4}
  {L : Type u_5} [IsDomain R] [IsDomain S] [inst_5 : Field K] [inst_6 : Field L] [inst_7 : Algebra R K]
  [inst_8 : Algebra R L] [inst_9 : Algebra S L] [Algebra.IsAlgebraic R S] [IsFractionRing S L] [inst_12 : Algebra K L]
  [IsScalarTower R S L] [IsScalarTower R K L] {a : S} {b : Set S},
  Function.Injective ⇑(algebraMap R L) →
    ↑(Ideal.span {a}) ⊆ ↑(Submodule.span R b) →
      ↑(Ideal.span {(algebraMap S L) a}) ⊆ ↑(Submodule.span K (⇑(algebraMap S L) '' b))

If the S-multiples of a are contained in some R-span, then Frac(S)-multiples of a are contained in the equivalent Frac(R)-span.

Defined in

Mathlib.RingTheory.Localization.Integral

Cited by

1 results in Mathlib

Foundations

Depth 123 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommRingCommRingAlgebraIsDomainIsDomainFieldFieldAlgebraAlgebraAlgebraAlgebra.IsAlgebraicIsFractionRingAlgebraIsScalarTowerIsScalarTower