Theorems · Theorem · commutative algebra
IsDiscreteValuationRing.irreducible_iff_uniformizer
∀ {R : Type u} [inst : CommRing R] [inst_1 : IsDomain R] [inst_2 : IsDiscreteValuationRing R] (ϖ : R),
Irreducible ϖ ↔ IsLocalRing.maximalIdeal R = Ideal.span {ϖ}An element of a DVR is irreducible iff it is a uniformizer, that is, generates the
maximal ideal of R.
- Cited by
- 4 results in Mathlib
- Foundations
- Depth 35 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites14
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Setstatement · cited by 53,352
- CommRingstatement and proof · cited by 17,173
- Idealstatement and proof · cited by 4,748
- Bot.botproof · cited by 4,720
- IsDomainstatement and proof · cited by 2,196
- Ideal.spanstatement and proof · cited by 948
- Irreduciblestatement and proof · cited by 496
- IsLocalRing.maximalIdealstatement and proof · cited by 297
- IsDiscreteValuationRingstatement and proof · cited by 117
- Ideal.span_singleton_eq_botproof · cited by 27
- IsLocalRing.eq_maximalIdealproof · cited by 20
- PrincipalIdealRing.isMaximal_of_irreducibleproof · cited by 4
Cited by4
Results whose statement or proof uses this declaration.
- Irreducible.maximalIdeal_eqproof · cited by 5
- IsDiscreteValuationRing.associated_of_irreducibleproof · cited by 2
- IsDiscreteValuationRing.iff_pid_with_one_nonzero_primeproof · cited by 2
- IsDiscreteValuationRing.exists_lift_of_le_oneproof · cited by 2