Theorems · Theorem · commutative algebra

Subalgebra.LinearDisjoint.linearIndependent_left_of_flat_of_commute

∀ {R : Type u} {S : Type v} [inst : CommRing R] [inst_1 : Ring S] [inst_2 : Algebra R S] {A B : Subalgebra R S},
  A.LinearDisjoint B →
    ∀ [Module.Flat R ↥B] {ι : Type u_1} {a : ι → ↥A},
      LinearIndependent R a → (∀ (a : ↥A) (b : ↥B), Commute ↑a ↑b) → LinearIndependent (↥B) (⇑A.val ∘ a)

If A and B are linearly disjoint and their elements commute, if B is a flat R-module, then for any family of R-linearly independent elements of A, they are also B-linearly independent.

Defined in

Mathlib.RingTheory.LinearDisjoint

Cited by

1 results in Mathlib

Foundations

Depth 105 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommRingRingAlgebraModule.Flat