Theorems · Theorem · commutative algebra
IsDiscreteValuationRing.iff_pid_with_one_nonzero_prime
∀ (R : Type u) [inst : CommRing R] [inst_1 : IsDomain R], IsDiscreteValuationRing R ↔ IsPrincipalIdealRing R ∧ ∃! P, P ≠ ⊥ ∧ P.IsPrime
An integral domain is a DVR iff it's a PID with a unique non-zero prime ideal.
- Cited by
- 2 results in Mathlib
- Foundations
- Depth 78 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites28
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
- Submoduleproof · cited by 7,192
- Idealstatement and proof · cited by 4,748
- Bot.botstatement and proof · cited by 4,720
- IsDomainstatement and proof · cited by 2,196
- le_antisymmproof · cited by 2,068
- le_reflproof · cited by 2,061
- Submodule.spanproof · cited by 1,504
- le_transproof · cited by 985
- Ideal.IsPrimestatement and proof · cited by 827
- Irreducibleproof · cited by 496
- le_of_eqproof · cited by 366
Cited by2
Results whose statement or proof uses this declaration.
- IsLocalization.OverPrime.mem_normalizedFactors_of_isPrimeproof · cited by 1
- IsDiscreteValuationRing.of_ufd_of_unique_irreducibleproof · cited by 1