Theorems · Theorem · commutative algebra
IsLocalization.injective
∀ {R : Type u_1} [inst : CommRing R] {M : Submonoid R} (S : Type u_2) [inst_1 : CommRing S] [inst_2 : Algebra R S]
[IsLocalization M S], M ≤ nonZeroDivisors R → Function.Injective ⇑(algebraMap R S)- Defined in
- Mathlib.RingTheory.Localization.Defs
- Cited by
- 22 results in Mathlib
- Foundations
- Depth 25 from the axioms · uses propext, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites10
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- DFunLike.coestatement · cited by 62,936
- CommRingstatement and proof · cited by 17,173
- Algebrastatement and proof · cited by 11,388
- RingHomstatement · cited by 10,189
- Algebra.algebraMapstatement · cited by 4,706
- Submonoidstatement and proof · cited by 3,086
- nonZeroDivisorsstatement and proof · cited by 895
- IsLocalizationstatement and proof · cited by 636
- isRegular_iff_mem_nonZeroDivisorsproof · cited by 6
- IsLocalization.injectiveₛproof · cited by 1
Cited by22
Results whose statement or proof uses this declaration.
- IsFractionRing.injectiveproof · cited by 70
- IsLocalization.rank_eqproof · cited by 7
- Algebra.norm_eq_iffproof · cited by 5
- FractionalIdeal.count_well_definedproof · cited by 5
- IsLocalization.coeSubmodule_le_coeSubmoduleproof · cited by 4
- IsField.localization_map_bijectiveproof · cited by 3
- Ideal.ramificationIdx'_eq_one_of_map_localizationproof · cited by 3
- Ideal.isPrincipal_of_isPrincipal_isLocalizationAway_of_primeproof · cited by 2
- IsLocalization.bot_lt_under_primeproof · cited by 2
- Polynomial.not_weaklyQuasiFiniteAtproof · cited by 2
- IsLocalization.AtPrime.not_isFieldproof · cited by 2
- IsAlmostIntegral.isIntegral_of_nonZeroDivisors_le_comapproof · cited by 1