Theorems · Theorem · commutative algebra

Submodule.LinearDisjoint.linearIndependent_right_of_flat

∀ {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 ↥M] {ι : Type u_1} {n : ι → ↥N}, LinearIndependent R n → (M.mulRightMap n).ker = ⊥

If M and N are linearly disjoint, if M is a flat R-module, then for any family of R-linearly independent elements { n_i } of N, they are also M-linearly independent, in the sense that the R-linear map from ι →₀ M to S which maps { m_i } to the sum of m_i * n_i (Submodule.mulRightMap M n) has trivial kernel.

Defined in

Mathlib.LinearAlgebra.LinearDisjoint

Cited by

2 results in Mathlib

Foundations

Depth 103 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommRingRingAlgebraModule.Flat