Theorems · Theorem · commutative algebra
Ideal.mem_minimalPrimes_of_height_eq
Deprecated since 2026-07-28Use Ideal.mem_minimalPrimes_of_height_le instead.
∀ {R : Type u_1} [inst : CommRing R] {I J : Ideal R},
I ≤ J → ∀ [J.IsPrime] [J.FiniteHeight], J.height ≤ I.height → J ∈ I.minimalPrimesAlias of Ideal.mem_minimalPrimes_of_height_le.
If J is a prime ideal containing I, and its height is less than or equal to the height of I,
then J is a minimal prime over I
- Defined in
- Mathlib.RingTheory.Ideal.Height
- Cited by
- 0 results in Mathlib
- Foundations
- Depth 74 from the axioms · uses propext, Classical.choice, Quot.sound
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Cites9
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Setstatement · cited by 53,352
- CommRingstatement · cited by 17,173
- ENatstatement · cited by 4,985
- Idealstatement · cited by 4,748
- Ideal.IsPrimestatement · cited by 827
- Ideal.heightstatement · cited by 83
- Ideal.minimalPrimesstatement · cited by 74
- Ideal.FiniteHeightstatement · cited by 18
- Ideal.mem_minimalPrimes_of_height_leproof · cited by 3
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