Theorems · Theorem · commutative algebra
Ideal.prime_iff_isPrime
∀ {A : Type u_2} [inst : CommRing A] [IsDedekindDomain A] {P : Ideal A}, P ≠ ⊥ → (Prime P ↔ P.IsPrime)In a Dedekind domain, the (nonzero) prime elements of the monoid with zero Ideal A
are exactly the prime ideals.
- Cited by
- 6 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.
Cites8
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
- Ideal.IsPrimestatement · cited by 827
- IsDedekindDomainstatement and proof · cited by 668
- Primestatement · cited by 277
- Ideal.isPrime_of_primeproof · cited by 17
- Ideal.prime_of_isPrimeproof · cited by 15
Cited by6
Results whose statement or proof uses this declaration.
- Ideal.prime_span_singleton_iffproof · cited by 5
- Ideal.mem_normalizedFactors_iffproof · cited by 3
- dvd_differentIdeal_of_not_isSeparableproof · cited by 1
- Ideal.map_prime_of_equivproof · cited by 1
- Ideal.singleton_span_mem_normalizedFactors_of_mem_normalizedFactorsproof · cited by 1
- Ideal.quotientToQuotientRangePowQuotSucc_surjectiveproof · cited by 0