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Theorems · Theorem · commutative algebra

Ideal.exists_ideal_lt_liesOver_of_lt

∀ {R : Type u_1} {S : Type u_2} [inst : CommRing R] [inst_1 : CommRing S] [inst_2 : Algebra R S]
  [Algebra.HasGoingDown R S] {p q : Ideal R} [p.IsPrime] [q.IsPrime] (Q : Ideal S) [Q.IsPrime] [Q.LiesOver q],
  p < q → ∃ P < Q, P.IsPrime ∧ P.LiesOver p
Defined in
Mathlib.RingTheory.Ideal.GoingDown
Cited by
1 results in Mathlib
Foundations
Depth 26 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes
CommRingCommRingAlgebraAlgebra.HasGoingDownIdeal.IsPrimeIdeal.IsPrimeIdeal.IsPrimeIdeal.LiesOver
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Around this declaration

Dashed lines are statement dependencies; solid lines are citations in proofs.

Ideal.exists_ltSeries_of_hasGoingDown · cited by 1Ideal.exists_ltSeries_of_…CommRing · cited by 17173CommRingAlgebra · cited by 11388AlgebraIdeal · cited by 4748IdealLT.lt.le · cited by 2189lt.leIdeal.IsPrime · cited by 827Ideal.IsPrimeIdeal.LiesOver · cited by 272Ideal.LiesOverIdeal.under · cited by 170Ideal.underIdeal.over_def · cited by 60Ideal.over_defeq_of_le_of_not_lt · cited by 28eq_of_le_of_not_ltAlgebra.HasGoingDown · cited by 11Algebra.HasGoingDownIdeal.exists_ideal_le_liesOver_of_le · cited by 2Ideal.exists_ideal_le_lie…Ideal.exists_ideal_lt_liesOve…CITED BYCITES

Cites11

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Cited by1

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