Theorems · Theorem · commutative algebra
FractionalIdeal.exists_eq_spanSingleton_mul
∀ {R₁ : Type u_3} [inst : CommRing R₁] {K : Type u_4} [inst_1 : Field K] [inst_2 : Algebra R₁ K]
[inst_3 : IsFractionRing R₁ K] [IsDomain R₁] (I : FractionalIdeal (nonZeroDivisors R₁) K),
∃ a aI, a ≠ 0 ∧ I = FractionalIdeal.spanSingleton (nonZeroDivisors R₁) ((algebraMap R₁ K) a)⁻¹ * ↑aI- Cited by
- 6 results in Mathlib
- Foundations
- Depth 81 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites28
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- DFunLike.coestatement and proof · cited by 62,936
- CommRingstatement and proof · cited by 17,173
- Algebrastatement and proof · cited by 11,388
- RingHomstatement · cited by 10,189
- Fieldstatement and proof · cited by 7,404
- Idealstatement · cited by 4,748
- Algebra.algebraMapstatement and proof · cited by 4,706
- one_mulproof · cited by 2,841
- IsDomainstatement and proof · cited by 2,196
- mul_assocproof · cited by 1,667
- nonZeroDivisorsstatement and proof · cited by 895
- IsFractionRingstatement and proof · cited by 738
Cited by6
Results whose statement or proof uses this declaration.
- FractionalIdeal.count_well_definedproof · cited by 5
- FractionalIdeal.count_mulproof · cited by 4
- FractionalIdeal.count_ne_zerostatement and proof · cited by 2
- FractionalIdeal.isNoetherianproof · cited by 1
- FractionalIdeal.finite_factorsproof · cited by 0
- FractionalIdeal.finprod_heightOneSpectrum_factorization'proof · cited by 0