Theorems · Theorem · commutative algebra
IsFractionRing.div_surjective
∀ (A : Type u_4) [inst : CommRing A] {K : Type u_5} [inst_1 : Field K] [inst_2 : Algebra A K] [IsFractionRing A K]
(z : K), ∃ x, ∃ y ∈ nonZeroDivisors A, (algebraMap A K) x / (algebraMap A K) y = z- Cited by
- 22 results in Mathlib
- Foundations
- Depth 47 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.
- 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
- Algebra.algebraMapstatement and proof · cited by 4,706
- Submonoidstatement · cited by 3,086
- nonZeroDivisorsstatement and proof · cited by 895
- IsFractionRingstatement and proof · cited by 738
- IsLocalization.mk'proof · cited by 218
- IsLocalization.exists_mk'_eqproof · cited by 63
- IsFractionRing.mk'_eq_divproof · cited by 18
Cited by22
Results whose statement or proof uses this declaration.
- IsGaloisGroup.of_isFractionRingproof · cited by 5
- Ideal.IsFractionRing.normalproof · cited by 3
- IsFractionRing.isInvariant_of_isIntegralproof · cited by 2
- ValuationRing.iff_isInteger_or_isIntegerproof · cited by 2
- IsDiscreteValuationRing.exists_lift_of_le_oneproof · cited by 2
- IsFractionRing.ringHom_extproof · cited by 2
- Ring.ordFrac_eq_inverse_comp_valuationproof · cited by 2
- IsDedekindDomain.HeightOneSpectrum.valuation_liesOverproof · cited by 1
- IsFractionRing.algHom_commutesproof · cited by 1
- IsFractionRing.closure_range_algebraMapproof · cited by 1
- NumberField.FinitePlace.hasFiniteMulSupportproof · cited by 1
- IsGaloisGroup.fixingSubgroup_range_algebraMapproof · cited by 1