Theorems · Theorem · commutative algebra
FractionalIdeal.spanSingleton_le_iff_mem
∀ {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 : P} {I : FractionalIdeal S P}, FractionalIdeal.spanSingleton S x ≤ I ↔ x ∈ I- Cited by
- 3 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.
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
- Submoduleproof · cited by 7,192
- Submonoidstatement and proof · cited by 3,086
- IsLocalizationstatement and proof · cited by 636
- FractionalIdealstatement and proof · cited by 423
- FractionalIdeal.coeToSubmoduleproof · cited by 130
- FractionalIdeal.spanSingletonstatement · cited by 73
- Submodule.span_singleton_le_iff_memproof · cited by 21
- FractionalIdeal.coe_spanSingletonproof · cited by 11
- FractionalIdeal.coe_le_coeproof · cited by 10
- FractionalIdeal.mem_coeproof · cited by 9
Cited by3
Results whose statement or proof uses this declaration.
- FractionalIdeal.dual_eq_dual_mul_dualproof · cited by 1
- FractionalIdeal.isPrincipal_of_unit_of_comap_mul_span_singleton_eq_topproof · cited by 1
- FractionalIdeal.num_le_mul_invproof · cited by 1