Theorems · Theorem · commutative algebra
Ring.krullDimLE_zero_iff
∀ {R : Type u_1} [inst : CommSemiring R], Ring.KrullDimLE 0 R ↔ ∀ (I : Ideal R), I.IsPrime → I.IsMaximal- Defined in
- Mathlib.RingTheory.KrullDimension.Basic
- Cited by
- 5 results in Mathlib
- Foundations
- Depth 77 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.
Cites12
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
- Idealstatement and proof · cited by 4,748
- Nat.cast_zeroproof · cited by 1,870
- Ideal.IsPrimestatement and proof · cited by 827
- PrimeSpectrumproof · cited by 625
- Ideal.IsMaximalstatement and proof · cited by 452
- IsMaxproof · cited by 372
- RelIso.toEquivproof · cited by 113
- Order.krullDimproof · cited by 82
- Ring.KrullDimLEstatement · cited by 79
- Equiv.forall_congr_leftproof · cited by 41
- PrimeSpectrum.equivSubtypeproof · cited by 8
Cited by5
Results whose statement or proof uses this declaration.
- Ring.KrullDimLE.mk₀proof · cited by 4
- Ring.krullDimLE_zero_iff_forall_minimalPrimes_isMaximalproof · cited by 1
- IsLocalRing.quotient_artinian_of_mem_minimalPrimes_of_isLocalRingproof · cited by 1
- Ideal.isMaximal_of_isPrimeproof · cited by 1