Theorems · Theorem · commutative algebra

Module.FaithfullyFlat.iff_flat_and_lTensor_faithful

∀ (R : Type u) (M : Type v) [inst : CommRing R] [inst_1 : AddCommGroup M] [inst_2 : Module R M],
  Module.FaithfullyFlat R M ↔
    Module.Flat R M ∧
      ∀ (N : Type (max u v)) [inst_3 : AddCommGroup N] [inst_4 : Module R N],
        Nontrivial N → Nontrivial (TensorProduct R M N)

Defined in: Mathlib.RingTheory.Flat.FaithfullyFlat.Basic
Cited by: 2 results in Mathlib
Foundations: Depth 104 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes: CommRing AddCommGroup Module

Mathlib docs Source Loogle LeanSearch

Around this declaration

Dashed lines are statement dependencies; solid lines are citations in proofs.

Cites12

Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.

Modulestatement and proof · cited by 20,661
CommRingstatement and proof · cited by 17,173
AddCommGroupstatement and proof · cited by 12,871
TensorProductstatement and proof · cited by 2,545
Nontrivialstatement and proof · cited by 2,416
LinearEquiv.symmproof · cited by 1,461
Module.Flatstatement and proof · cited by 279
TensorProduct.commproof · cited by 108
LinearEquiv.toEquivproof · cited by 105
Module.FaithfullyFlatstatement · cited by 72
Equiv.nontrivialproof · cited by 8
Module.FaithfullyFlat.iff_flat_and_rTensor_faithfulproof · cited by 2

Cited by2

Results whose statement or proof uses this declaration.

Module.FaithfullyFlat.of_linearEquivproof · cited by 1
Module.FaithfullyFlat.iff_flat_and_lTensor_reflects_trivialityproof · cited by 1