Theorems · Theorem · commutative algebra
PrimeSpectrum.subsingleton_iff_isField_of_isReduced
∀ {R : Type u_2} [inst : CommRing R] [IsReduced R] [Nontrivial R], Subsingleton (PrimeSpectrum R) ↔ IsField R- Defined in
- Mathlib.RingTheory.KrullDimension.Zero
- Cited by
- 1 results in Mathlib
- Foundations
- Depth 90 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- CommRingIsReducedNontrivial
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites11
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
- Nontrivialstatement and proof · cited by 2,416
- PrimeSpectrumstatement and proof · cited by 625
- IsLocalRingproof · cited by 339
- IsFieldstatement and proof · cited by 103
- IsReducedstatement and proof · cited by 98
- MaximalSpectrumproof · cited by 73
- Function.Injective.subsingletonproof · cited by 16
- Ring.KrullDimLE.isField_of_isReducedproof · cited by 2
- IsLocalRing.of_singleton_maximalSpectrumproof · cited by 2
- MaximalSpectrum.toPrimeSpectrum_injectiveproof · cited by 1
Cited by1
Results whose statement or proof uses this declaration.
- AlgebraicGeometry.isField_stalk_of_closure_mem_irreducibleComponentsproof · cited by 1