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feat(RingTheory/Ideal/GoingUp): a prime ideal is minimal over the image of its preimage #39672

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feat(RingTheory/Ideal/GoingUp): a prime ideal is minimal over the image of its preimage #39672

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6 changes: 6 additions & 0 deletions Mathlib/RingTheory/Ideal/GoingUp.lean
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Expand Up @@ -233,6 +233,12 @@ theorem IsIntegral.isMaximal_of_isMaximal_comap (I : Ideal A) [I.IsPrime]
@[deprecated (since := "2026-05-08")] alias IsIntegralClosure.isMaximal_of_isMaximal_comap :=
IsIntegral.isMaximal_of_isMaximal_comap

theorem IsIntegral.mem_minimalPrimes_map_under (I : Ideal A) [I.IsPrime] :
I ∈ ((I.under R).map (algebraMap R A)).minimalPrimes := by
refine ⟨⟨inferInstance, map_comap_le⟩, fun r ⟨hr, hpr⟩ hrq ↦ ?_⟩
contrapose! hpr
exact mt map_le_iff_le_comap.mp (not_le_of_gt (IsIntegral.comap_lt_comap (hrq.lt_of_not_ge hpr)))

variable [IsDomain A]

variable (R) in
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