Theorems · Theorem · commutative algebra

IsIntegral.tower_bot

∀ {R : Type u_1} {A : Type u_2} {B : Type u_3} [inst : CommRing R] [inst_1 : CommRing A] [inst_2 : Ring B]
  [inst_3 : Algebra A B] [inst_4 : Algebra R B] [inst_5 : Algebra R A] [IsScalarTower R A B],
  Function.Injective ⇑(algebraMap A B) → ∀ {x : A}, IsIntegral R ((algebraMap A B) x) → IsIntegral R x

If R → A → B is an algebra tower with A → B injective, then if the entire tower is an integral extension so is R → A

Defined in

Mathlib.RingTheory.IntegralClosure.IsIntegralClosure.Basic

Cited by

3 results in Mathlib

Foundations

Depth 114 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommRingCommRingRingAlgebraAlgebraAlgebraIsScalarTower