Theorems · Theorem · commutative algebra
Module.Flat.iff_lTensor_injective
∀ {R : Type u} {M : Type v} [inst : CommRing R] [inst_1 : AddCommGroup M] [inst_2 : Module R M],
Module.Flat R M ↔ ∀ ⦃I : Ideal R⦄, I.FG → Function.Injective ⇑(LinearMap.lTensor M (Submodule.subtype I))The lTensor-variant of iff_rTensor_injective.
- Defined in
- Mathlib.RingTheory.Flat.Tensor
- Cited by
- 1 results in Mathlib
- Foundations
- Depth 110 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- CommRingAddCommGroupModule
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites16
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- DFunLike.coestatement and proof · cited by 62,936
- Modulestatement and proof · cited by 20,661
- RingHom.idstatement · cited by 18,349
- CommRingstatement and proof · cited by 17,173
- AddCommGroupstatement and proof · cited by 12,871
- LinearMapstatement · cited by 10,215
- Submodulestatement · cited by 7,192
- Idealstatement and proof · cited by 4,748
- TensorProductstatement · cited by 2,545
- Submodule.subtypestatement and proof · cited by 480
- Module.Flatstatement and proof · cited by 279
- LinearMap.rTensorproof · cited by 266
Cited by1
Results whose statement or proof uses this declaration.
- Ideal.Pure.of_inf_eq_mulproof · cited by 0