Theorems · Theorem · algebraic geometry
PrimeSpectrum.isMax_iff
∀ {R : Type u} [inst : CommSemiring R] {x : PrimeSpectrum R}, IsMax x ↔ x.asIdeal.IsMaximalAlso see PrimeSpectrum.isClosed_singleton_iff_isMaximal
- Defined in
- Mathlib.RingTheory.Spectrum.Prime.Basic
- Cited by
- 1 results in Mathlib
- Foundations
- Depth 75 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- CommSemiring
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites14
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- CommSemiringstatement and proof · cited by 10,911
- Top.topproof · cited by 9,680
- Idealproof · cited by 4,748
- LT.lt.trans_leproof · cited by 678
- PrimeSpectrumstatement and proof · cited by 625
- Ideal.IsMaximalstatement and proof · cited by 452
- Eq.geproof · cited by 375
- IsMaxstatement and proof · cited by 372
- PrimeSpectrum.asIdealstatement and proof · cited by 333
- Ideal.IsPrime.ne_topproof · cited by 82
- Ideal.IsMaximal.isPrimeproof · cited by 53
- Ideal.exists_le_maximalproof · cited by 47
Cited by1
Results whose statement or proof uses this declaration.
- PrimeSpectrum.isOpen_singleton_tfae_of_isNoetherian_of_isJacobsonRingproof · cited by 2