Theorems · Theorem · algebraic geometry
PrimeSpectrum.exists_primeSpectrum_prod_le
∀ (R : Type u) [inst : CommRing R] [IsNoetherianRing R] (I : Ideal R), ∃ Z, (Multiset.map PrimeSpectrum.asIdeal Z).prod ≤ I
In a Noetherian ring, every ideal contains a product of prime ideals ([samuel1967, § 3.3, Lemma 3]).
- Defined in
- Mathlib.RingTheory.Spectrum.Prime.Basic
- Cited by
- 0 results in Mathlib
- Foundations
- Depth 85 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- CommRingIsNoetherianRing
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Cites35
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Setproof · cited by 53,352
- CommRingstatement and proof · cited by 17,173
- Top.topproof · cited by 9,680
- Submoduleproof · cited by 7,192
- Idealstatement and proof · cited by 4,748
- Multisetstatement and proof · cited by 2,627
- le_reflproof · cited by 2,061
- Submodule.spanproof · cited by 1,504
- le_transproof · cited by 985
- Multiset.mapstatement and proof · cited by 876
- Ideal.IsPrimeproof · cited by 827
- PrimeSpectrumstatement and proof · cited by 625
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