Theorems · Theorem · commutative algebra

TensorProduct.exists_finite_submodule_left_of_setFinite

∀ {R : Type u_1} {M : Type u_2} {N : Type u_3} [inst : CommSemiring R] [inst_1 : AddCommMonoid M]
  [inst_2 : AddCommMonoid N] [inst_3 : Module R M] [inst_4 : Module R N] (s : Set (TensorProduct R M N)),
  s.Finite → ∃ M', Module.Finite R ↥M' ∧ s ⊆ ↑(LinearMap.rTensor N M'.subtype).range

For a finite subset s of M ⊗[R] N, there exists a finitely generated submodule M' of M, such that s is contained in the image of M' ⊗[R] N in M ⊗[R] N.

Defined in

Mathlib.LinearAlgebra.TensorProduct.Finiteness

Cited by

0 results in Mathlib

Foundations

Depth 86 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommSemiringAddCommMonoidAddCommMonoidModuleModule