quotQuotEquivQuotSup — Mathlib

English

The kernel of quotQuotMk coincides with the maximal of I and J.

Русский

Ядро quotQuotMk совпадает с максимумом I и J.

LaTeX

$$$\\ker(\\mathrm{quotQuotMk}\\; I\\; J) = \\mathrm{Max}(I,J)$$$

Lean4

/-- `quotQuotToQuotSup` and `liftSupQuotQuotMk` are inverse isomorphisms. In the case where
`I ≤ J`, this is the Third Isomorphism Theorem (see `quotQuotEquivQuotOfLe`). -/
def quotQuotEquivQuotSup : (R ⧸ I) ⧸ J.map (Ideal.Quotient.mk I) ≃+* R ⧸ I ⊔ J :=
  RingEquiv.ofHomInv (quotQuotToQuotSup I J) (liftSupQuotQuotMk I J)
    (by
      repeat apply Ideal.Quotient.ringHom_ext
      rfl)
    (by
      repeat apply Ideal.Quotient.ringHom_ext
      rfl)

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