Theorems · Theorem · commutative algebra

Ideal.spanNorm_spanNorm_of_bot_or_top

∀ (R : Type u_1) [inst : CommRing R] [inst_1 : IsDomain R] {S : Type u_3} [inst_2 : CommRing S] [inst_3 : IsDomain S]
  [inst_4 : IsIntegrallyClosed R] [inst_5 : IsIntegrallyClosed S] [inst_6 : Algebra R S] [inst_7 : Module.Finite R S]
  [inst_8 : Module.IsTorsionFree R S] (T : Type u_4) [inst_9 : CommRing T] [inst_10 : IsDomain T]
  [inst_11 : IsIntegrallyClosed T] [inst_12 : Algebra R T] [inst_13 : Algebra T S] [inst_14 : Module.Finite R T]
  [inst_15 : Module.Finite T S] [inst_16 : Module.IsTorsionFree R T] [inst_17 : Module.IsTorsionFree T S]
  [IsScalarTower R T S],
  (∀ (I : Ideal R), I = ⊥ ∨ I = ⊤) → ∀ (I : Ideal S), Ideal.spanNorm R (Ideal.spanNorm T I) = Ideal.spanNorm R I

This condition eq_bot_or_top is equivalent to being a field. However, Ideal.spanNorm_spanNorm_of_field would be harder to apply since we'd need to upgrade a CommRing R instance to a Field R instance.

Defined in

Mathlib.RingTheory.Ideal.Norm.RelNorm

Cited by

1 results in Mathlib

Foundations

Depth 188 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommRingIsDomainCommRingIsDomainIsIntegrallyClosedIsIntegrallyClosedAlgebraModule.FiniteModule.IsTorsionFreeCommRingIsDomainIsIntegrallyClosedAlgebraAlgebraModule.FiniteModule.FiniteModule.IsTorsionFreeModule.IsTorsionFreeIsScalarTower