Theorems · Theorem · commutative algebra

Submodule.LinearDisjoint.linearIndependent_mul_of_flat_right

∀ {R : Type u} {S : Type v} [inst : CommRing R] [inst_1 : Ring S] [inst_2 : Algebra R S] {M N : Submodule R S},
  M.LinearDisjoint N →
    ∀ [Module.Flat R ↥N] {κ : Type u_1} {ι : Type u_2} {m : κ → ↥M} {n : ι → ↥N},
      LinearIndependent R m → LinearIndependent R n → LinearIndependent R fun i => ↑(m i.1) * ↑(n i.2)

If M and N are linearly disjoint, if N is flat, then for any family of R-linearly independent elements { m_i } of M, and any family of R-linearly independent elements { n_j } of N, the family { m_i * n_j } in S is also R-linearly independent.

Defined in

Mathlib.LinearAlgebra.LinearDisjoint

Cited by

2 results in Mathlib

Foundations

Depth 107 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommRingRingAlgebraModule.Flat