Theorems · Theorem · commutative algebra

Algebra.HasGoingUp.exists_ideal_ge_liesOver_of_lt

∀ {R : Type u_1} {S : Type u_2} {inst : CommRing R} {inst_1 : CommRing S} {inst_2 : Algebra R S}
  [self : Algebra.HasGoingUp R S] {q : Ideal R} [q.IsPrime] (P : Ideal S) [P.IsPrime],
  Ideal.under R P < q → ∃ Q, P ≤ Q ∧ Q.IsPrime ∧ Q.LiesOver q

Defined in: Mathlib.RingTheory.Ideal.HasGoingUp
Cited by: 1 results in Mathlib
Foundations: Depth 24 from the axioms · uses propext, Quot.sound
Assumes: Algebra.HasGoingUp Ideal.IsPrime Ideal.IsPrime

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Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.

CommRingstatement and proof · cited by 17,173
Algebrastatement and proof · cited by 11,388
Idealstatement · cited by 4,748
Ideal.IsPrimestatement · cited by 827
Ideal.LiesOverstatement · cited by 272
Ideal.understatement · cited by 170
Algebra.HasGoingUpstatement and proof · cited by 6

Cited by1

Results whose statement or proof uses this declaration.

Ideal.exists_ideal_ge_liesOver_of_leproof · cited by 2