Theorems · Definition · commutative algebra

Ideal.quotientEquivAlg

{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 : Ring B] →
              [inst_4 : Algebra R₁ B] →
                (I : Ideal A) →
                  (J : Ideal B) →
                    [inst_5 : I.IsTwoSided] →
                      [inst_6 : J.IsTwoSided] → (f : A ≃ₐ[R₁] B) → J = Ideal.map (↑f) I → (A ⧸ I) ≃ₐ[R₁] B ⧸ J

The algebra equiv A/I ≃ₐ[R] B/J induced by an algebra equiv f : A ≃ₐ[R] B, where J = f(I).

Defined in

Mathlib.RingTheory.Ideal.Quotient.Operations

Cited by

7 results in Mathlib

Foundations

Depth 92 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommSemiringRingAlgebraRingAlgebraIdeal.IsTwoSidedIdeal.IsTwoSided