Theorems · Theorem · commutative algebra

IsIntegral.pair_iff

∀ {R : Type u_1} {A : Type u_2} {B : Type u_3} [inst : CommRing R] [inst_1 : Ring A] [inst_2 : Ring B]
  [inst_3 : Algebra R A] [inst_4 : Algebra R B] {x : A × B}, IsIntegral R x ↔ IsIntegral R x.1 ∧ IsIntegral R x.2

An element of a product algebra is integral iff each component is integral.

Defined in

Mathlib.RingTheory.IntegralClosure.IsIntegral.Basic

Cited by

0 results in Mathlib

Foundations

Depth 115 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommRingRingRingAlgebraAlgebra