Theorems · Theorem · commutative algebra
FractionalIdeal.eq_spanSingleton_of_principal
∀ {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) [inst_4 : (↑I).IsPrincipal],
I = FractionalIdeal.spanSingleton S (Submodule.IsPrincipal.generator ↑I)- Cited by
- 3 results in Mathlib
- Foundations
- Depth 33 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites16
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
- Submoduleproof · cited by 7,192
- Submonoidstatement and proof · cited by 3,086
- Submodule.spanproof · cited by 1,504
- 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 · cited by 73
- Submodule.IsPrincipal.generatorstatement and proof · cited by 56
- Submodule.IsPrincipal.span_singleton_generatorproof · cited by 10
Cited by3
Results whose statement or proof uses this declaration.
- FractionalIdeal.mul_generator_self_invproof · cited by 2
- FractionalIdeal.isPrincipal_iffproof · cited by 1
- FractionalIdeal.invertible_iff_generator_nonzeroproof · cited by 0