Theorems · Theorem · commutative algebra
Ideal.isPrime_iff_bot_or_prime
∀ {A : Type u_2} [inst : CommRing A] [IsDedekindDomain A] {P : Ideal A}, P.IsPrime ↔ P = ⊥ ∨ Prime PIn a Dedekind domain, the prime ideals are the zero ideal together with the prime elements
of the monoid with zero Ideal A.
- Cited by
- 1 results in Mathlib
- Foundations
- Depth 148 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- CommRingIsDedekindDomain
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites9
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
- Idealstatement and proof · cited by 4,748
- Bot.botstatement and proof · cited by 4,720
- eq_or_neproof · cited by 1,117
- Ideal.IsPrimestatement and proof · cited by 827
- IsDedekindDomainstatement and proof · cited by 668
- Primestatement and proof · cited by 277
- Ideal.isPrime_of_primeproof · cited by 17
- Ideal.prime_of_isPrimeproof · cited by 15
Cited by1
Results whose statement or proof uses this declaration.
- Ideal.count_normalizedFactors_eqproof · cited by 3