Theorems · Theorem · commutative algebra
FractionalIdeal.spanSingleton_mul_spanSingleton
∀ {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] (x y : P),
FractionalIdeal.spanSingleton S x * FractionalIdeal.spanSingleton S y = FractionalIdeal.spanSingleton S (x * y)- Cited by
- 11 results in Mathlib
- Foundations
- Depth 80 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites12
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
- Submodule.spanproof · cited by 1,504
- IsLocalizationstatement and proof · cited by 636
- FractionalIdealstatement · cited by 423
- FractionalIdeal.spanSingletonstatement and proof · cited by 73
- FractionalIdeal.coeToSubmodule_injectiveproof · cited by 19
- Submodule.span_mul_spanproof · cited by 18
- FractionalIdeal.coe_mulproof · cited by 15
- FractionalIdeal.coe_spanSingletonproof · cited by 11
- Set.singleton_mul_singletonproof · cited by 7
Cited by11
Results whose statement or proof uses this declaration.
- FractionalIdeal.div_spanSingletonproof · cited by 4
- FractionalIdeal.count_mulproof · cited by 4
- FractionalIdeal.mk'_mul_coeIdeal_eq_coeIdealproof · cited by 3
- FractionalIdeal.spanSingleton_mul_invproof · cited by 2
- FractionalIdeal.mul_generator_self_invproof · cited by 2
- FractionalIdeal.one_div_spanSingletonproof · cited by 2
- FractionalIdeal.isNoetherian_spanSingleton_inv_to_map_mulproof · cited by 1
- IsDedekindDomain.exists_add_spanSingleton_mul_eqproof · cited by 1
- FractionalIdeal.spanSingleton_div_spanSingletonproof · cited by 1
- FractionalIdeal.spanSingleton_powproof · cited by 0