Theorems · Theorem · commutative algebra

FractionalIdeal.extended_spanSingleton

∀ {A : Type u_1} [inst : CommRing A] {B : Type u_2} [inst_1 : CommRing B] {f : A →+* B} {K : Type u_3} {M : Submonoid A}
  [inst_2 : CommRing K] [inst_3 : Algebra A K] [inst_4 : IsLocalization M K] (L : Type u_4) {N : Submonoid B}
  [inst_5 : CommRing L] [inst_6 : Algebra B L] [inst_7 : IsLocalization N L] (hf : M ≤ Submonoid.comap f N) (x : K),
  FractionalIdeal.extended L hf (FractionalIdeal.spanSingleton M x) =
    FractionalIdeal.spanSingleton N ((IsLocalization.map L f hf) x)

The extension of a principal fractional ideal is the principal ideal generated by the extension of a generator.

Defined in

Mathlib.RingTheory.FractionalIdeal.Extended

Cited by

1 results in Mathlib

Foundations

Depth 61 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommRingCommRingCommRingAlgebraIsLocalizationCommRingAlgebraIsLocalization