Theorems · Theorem · commutative algebra

Subalgebra.LinearDisjoint.linearIndependent_left_of_flat

∀ {R : Type u} {S : Type v} [inst : CommRing R] [inst_1 : CommRing 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 → LinearIndependent (↥B) (⇑A.val ∘ a)

In a commutative ring, if A and B are linearly disjoint, 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

2 results in Mathlib

Foundations

Depth 106 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommRingCommRingAlgebraModule.Flat