Theorems · Theorem · commutative algebra
IsLocalization.AtPrime.isUnit_to_map_iff
∀ {R : Type u_1} [inst : CommSemiring R] (S : Type u_2) [inst_1 : CommSemiring S] [inst_2 : Algebra R S] (I : Ideal R)
[hI : I.IsPrime] [IsLocalization.AtPrime S I] (x : R), IsUnit ((algebraMap R S) x) ↔ x ∈ I.primeCompl- Cited by
- 4 results in Mathlib
- Foundations
- Depth 83 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites19
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
- Algebrastatement and proof · cited by 11,388
- CommSemiringstatement and proof · cited by 10,911
- RingHomstatement · cited by 10,189
- SetLike.coeproof · cited by 8,199
- Idealstatement and proof · cited by 4,748
- Algebra.algebraMapstatement and proof · cited by 4,706
- Submonoidstatement · cited by 3,086
- IsUnitstatement and proof · cited by 1,602
- Ideal.IsPrimestatement and proof · cited by 827
- Ideal.mapproof · cited by 692
- Ideal.primeComplstatement and proof · cited by 462
Cited by4
Results whose statement or proof uses this declaration.
- IsLocalization.AtPrime.to_map_mem_maximal_iffproof · cited by 9
- AlgebraicGeometry.basicOpen_eq_of_affineproof · cited by 5
- Algebra.QuasiFiniteAt.of_isOpen_singletonproof · cited by 1
- ValuationSubring.ofPrime_valuation_eq_one_iff_mem_primeComplproof · cited by 0