Theorems · Theorem · commutative algebra
IsLocalization.to_map_ne_zero_of_mem_nonZeroDivisors
∀ {R : Type u_1} [inst : CommSemiring R] {M : Submonoid R} (S : Type u_2) [inst_1 : CommSemiring S]
[inst_2 : Algebra R S] [IsLocalization M S] [Nontrivial R],
M ≤ nonZeroDivisors R → ∀ {x : R}, x ∈ nonZeroDivisors R → (algebraMap R S) x ≠ 0- Defined in
- Mathlib.RingTheory.Localization.Defs
- Cited by
- 3 results in Mathlib
- Foundations
- Depth 26 from the axioms · uses propext, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites11
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
- Algebra.algebraMapstatement and proof · cited by 4,706
- Submonoidstatement and proof · cited by 3,086
- Nontrivialstatement and proof · cited by 2,416
- nonZeroDivisorsstatement and proof · cited by 895
- IsLocalizationstatement and proof · cited by 636
- nonZeroDivisors.ne_zeroproof · cited by 22
- IsLocalization.to_map_eq_zero_iffproof · cited by 2
Cited by3
Results whose statement or proof uses this declaration.
- IsFractionRing.to_map_ne_zero_of_mem_nonZeroDivisorsproof · cited by 10
- IsLocalization.OverPrime.mem_normalizedFactors_of_isPrimeproof · cited by 1
- exists_reduced_fraction'proof · cited by 1