Theorems · Theorem · commutative algebra
Ideal.isPrime_iff_of_isPrincipalIdealRing_of_noZeroDivisors
∀ {α : Type u} [inst : CommSemiring α] [IsPrincipalIdealRing α] [NoZeroDivisors α] [Nontrivial α] {P : Ideal α},
P.IsPrime ↔ P = ⊥ ∨ ∃ p, Prime p ∧ P = Ideal.span {p}- Defined in
- Mathlib.RingTheory.Ideal.Maximal
- Cited by
- 1 results in Mathlib
- Foundations
- Depth 34 from the axioms · uses propext, Classical.choice, Quot.sound
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.
- Setstatement · cited by 53,352
- 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.spanstatement · cited by 948
- Ideal.IsPrimestatement and proof · cited by 827
- NoZeroDivisorsstatement and proof · cited by 545
- Primestatement · cited by 277
- IsPrincipalIdealRingstatement and proof · cited by 131
- Ideal.isPrime_iff_of_isPrincipalIdealRingproof · cited by 1
Cited by1
Results whose statement or proof uses this declaration.
- Ideal.isPrime_int_iffproof · cited by 0