Theorems · Theorem · commutative algebra
FractionalIdeal.isPrincipal_iff
∀ {R : Type u_1} [inst : CommRing R] {S : Submonoid R} {P : Type u_2} [inst_1 : CommRing P] [inst_2 : Algebra R P]
[inst_3 : IsLocalization S P] (I : FractionalIdeal S P), (↑I).IsPrincipal ↔ ∃ x, I = FractionalIdeal.spanSingleton S x- 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.
- CommRingstatement and proof · cited by 17,173
- Algebrastatement and proof · cited by 11,388
- Submonoidstatement and proof · cited by 3,086
- IsLocalizationstatement and proof · cited by 636
- FractionalIdealstatement and proof · cited by 423
- FractionalIdeal.coeToSubmodulestatement and proof · cited by 130
- Submodule.IsPrincipalstatement and proof · cited by 129
- FractionalIdeal.spanSingletonstatement and proof · cited by 73
- Submodule.IsPrincipal.generatorproof · cited by 56
- FractionalIdeal.coe_spanSingletonproof · cited by 11
- FractionalIdeal.eq_spanSingleton_of_principalproof · cited by 3
Cited by1
Results whose statement or proof uses this declaration.
- FractionalIdeal.isPrincipal_invproof · cited by 0