Theorems · Theorem · commutative algebra

Subalgebra.LinearDisjoint.linearIndependent_right_of_flat

∀ {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 ↥A] {ι : Type u_1} {b : ι → ↥B}, LinearIndependent R b → LinearIndependent (↥A) (⇑B.val ∘ b)

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

Defined in

Mathlib.RingTheory.LinearDisjoint

Cited by

3 results in Mathlib

Foundations

Depth 104 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommRingRingAlgebraModule.Flat