Theorems · Theorem · commutative algebra
Module.Flat.iff_characterModule_injective
∀ {R : Type u} {M : Type v} [inst : CommRing R] [inst_1 : AddCommGroup M] [inst_2 : Module R M] [Small.{v, u} R],
Module.Flat R M ↔ Module.Injective R (CharacterModule M)CharacterModule M is an injective module iff M is flat.
See [Lambek_1964] for a self-contained proof.
- Defined in
- Mathlib.RingTheory.Flat.Tensor
- Cited by
- 1 results in Mathlib
- Foundations
- Depth 104 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites13
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- DFunLike.coeproof · cited by 62,936
- Modulestatement and proof · cited by 20,661
- RingHom.idproof · cited by 18,349
- CommRingstatement and proof · cited by 17,173
- AddCommGroupstatement and proof · cited by 12,871
- LinearMapproof · cited by 10,215
- Smallstatement and proof · cited by 369
- Module.Flatstatement and proof · cited by 279
- LinearMap.rTensorproof · cited by 266
- CharacterModulestatement · cited by 26
- Module.Injectivestatement · cited by 20
- Module.Flat.iff_rTensor_preserves_injective_linearMap'proof · cited by 3
Cited by1
Results whose statement or proof uses this declaration.
- Module.Flat.iff_characterModule_baerproof · cited by 0