Theorems · Theorem · commutative algebra

isIntegral_trans

∀ {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] [Algebra.IsIntegral R A]
  (x : B), IsIntegral A x → IsIntegral R x

If A is an R-algebra all of whose elements are integral over R, and x is an element of an A-algebra that is integral over A, then x is integral over R.

Defined in

Mathlib.RingTheory.IntegralClosure.IsIntegralClosure.Basic

Cited by

15 results in Mathlib

Foundations

Depth 132 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommRingCommRingRingAlgebraAlgebraAlgebraIsScalarTowerAlgebra.IsIntegral