Theorems · Theorem · commutative algebra
Ideal.kerLiftAlg_injective
∀ {R₁ : Type u_1} {A : Type u_3} {B : Type u_4} [inst : CommSemiring R₁] [inst_1 : Ring A] [inst_2 : Algebra R₁ A]
[inst_3 : Semiring B] [inst_4 : Algebra R₁ B] (f : A →ₐ[R₁] B), Function.Injective ⇑(Ideal.kerLiftAlg f)The induced algebra morphism from the quotient by the kernel is injective.
- Cited by
- 1 results in Mathlib
- Foundations
- Depth 89 from the axioms · uses propext, Classical.choice, Quot.sound
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.
- DFunLike.coestatement · cited by 62,936
- Semiringstatement and proof · cited by 13,802
- Algebrastatement and proof · cited by 11,388
- CommSemiringstatement and proof · cited by 10,911
- Ringstatement and proof · cited by 7,463
- Idealstatement · cited by 4,748
- AlgHomstatement and proof · cited by 3,236
- HasQuotient.Quotientstatement · cited by 2,301
- RingHomClass.toRingHomproof · cited by 746
- RingHom.kerstatement · cited by 363
- RingHom.kerLift_injectiveproof · cited by 5
- Ideal.kerLiftAlgstatement · cited by 4
Cited by1
Results whose statement or proof uses this declaration.
- exists_integral_inj_algHom_of_quotientproof · cited by 1