Theorems · Theorem · commutative algebra
Algebra.injective_lift_lsmul
∀ (R : Type u_1) (A : Type u_2) [inst : CommSemiring R] [inst_1 : CommSemiring A] [inst_2 : Algebra R A] [Algebra.IsEpi R A] (M : Type u_3) [inst_4 : AddCommMonoid M] [inst_5 : Module R M] [inst_6 : Module A M] [inst_7 : IsScalarTower R A M], Function.Injective ⇑(TensorProduct.lift (LinearMap.restrictScalars₁₂ R R (LinearMap.lsmul A M)))
If an R-algebra A is epi, then the scalar multiplication A ⊗[R] M → M is injective, for
any A-module M.
- Defined in
- Mathlib.Algebra.Algebra.Epi
- Cited by
- 1 results in Mathlib
- Foundations
- Depth 71 from the axioms · uses propext, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites29
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 and proof · cited by 18,349
- AddCommMonoidstatement and proof · cited by 12,281
- Algebrastatement and proof · cited by 11,388
- CommSemiringstatement and proof · cited by 10,911
- LinearMapstatement and proof · cited by 10,215
- IsScalarTowerstatement and proof · cited by 3,896
- TensorProductstatement and proof · cited by 2,545
- LinearMap.compproof · cited by 1,642
- one_smulproof · cited by 1,374
- TensorProduct.tmulproof · cited by 1,182
Cited by1
Results whose statement or proof uses this declaration.
- Submodule.injective_tensorToSpanproof · cited by 0