Theorems · Theorem · commutative algebra
Ring.not_isField_iff_exists_prime
∀ {R : Type u_5} [inst : CommSemiring R] [Nontrivial R], ¬IsField R ↔ ∃ p, p ≠ ⊥ ∧ p.IsPrime- Defined in
- Mathlib.RingTheory.Ideal.Basic
- Cited by
- 4 results in Mathlib
- Foundations
- Depth 78 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- CommSemiringNontrivial
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.
- CommSemiringstatement and proof · cited by 10,911
- Idealstatement and proof · cited by 4,748
- Bot.botstatement and proof · cited by 4,720
- Nontrivialstatement and proof · cited by 2,416
- Ideal.IsPrimestatement and proof · cited by 827
- Ideal.IsMaximalproof · cited by 452
- IsFieldstatement and proof · cited by 103
- Ideal.IsPrime.ne_topproof · cited by 82
- Ideal.IsMaximal.isPrimeproof · cited by 53
- Ring.not_isField_of_ne_of_neproof · cited by 2
- Ring.exists_maximal_of_not_isFieldproof · cited by 2
Cited by4
Results whose statement or proof uses this declaration.
- IsLocalRing.isField_iff_maximalIdeal_eqproof · cited by 8
- Ring.KrullDimLE.isField_of_isDomainproof · cited by 3
- PrimeSpectrum.t1Space_iff_isFieldproof · cited by 2
- PrimeSpectrum.exists_primeSpectrum_prod_le_and_ne_bot_of_domainproof · cited by 1