Theorems · Theorem · commutative algebra
IsPrincipalIdealRing.of_finite_maximals
∀ {R : Type u_1} [inst : CommRing R] [IsDedekindDomain R], {I | I.IsMaximal}.Finite → IsPrincipalIdealRing RA Dedekind domain is a PID if its set of maximal ideals is finite.
- Defined in
- Mathlib.RingTheory.DedekindDomain.PID
- Cited by
- 2 results in Mathlib
- Foundations
- Depth 140 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.
Cites13
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
- Set.ofPredstatement and proof · cited by 6,101
- Idealstatement and proof · cited by 4,748
- Bot.botproof · cited by 4,720
- Set.Finitestatement and proof · cited by 1,814
- eq_or_neproof · cited by 1,117
- IsDedekindDomainstatement and proof · cited by 668
- Ideal.IsMaximalstatement and proof · cited by 452
- IsPrincipalIdealRingstatement · cited by 131
- FractionalIdeal.coeIdealproof · cited by 109
- IsUnit.of_mul_eq_oneproof · cited by 43
- Ideal.IsPrincipal.of_finite_maximals_of_isUnitproof · cited by 1
Cited by2
Results whose statement or proof uses this declaration.
- IsPrincipalIdealRing.of_finite_primesproof · cited by 1
- isPrincipalIdealRing_of_isPrincipalIdealRing_isLocalization_maximalproof · cited by 0