Theorems · Theorem · ring theory
RingHom.injective
∀ {R : Type u_2} {S : Type u_3} [inst : NonAssocRing R] [IsSimpleRing R] [inst_2 : NonAssocSemiring S] [Nontrivial S]
(f : R →+* S), Function.Injective ⇑f- Defined in
- Mathlib.RingTheory.SimpleRing.Basic
- Cited by
- 187 results in Mathlib
- Foundations
- Depth 53 from the axioms, rests on 818 definitions · uses propext, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites8
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- DFunLike.coestatement · cited by 62,936
- RingHomstatement and proof · cited by 10,189
- Nontrivialstatement and proof · cited by 2,416
- NonAssocSemiringstatement and proof · cited by 805
- NonAssocRingstatement and proof · cited by 483
- IsSimpleRingstatement and proof · cited by 39
- not_subsingletonproof · cited by 24
- IsSimpleRing.injective_ringHom_or_subsingleton_codomainproof · cited by 2
Cited by189
Results whose statement or proof uses this declaration.
- Polynomial.natDegree_mapproof · cited by 21
- Polynomial.map_dvd_map'proof · cited by 14
- Polynomial.leadingCoeff_mapproof · cited by 11
- Polynomial.degree_mapproof · cited by 10
- Polynomial.Splits.roots_mapproof · cited by 10
- Algebra.isIntegral_traceproof · cited by 7
- IsAlgClosed.algebraMap_bijective_of_isIntegralproof · cited by 6
- minpoly.eq_X_sub_Cproof · cited by 6
- NumberField.mixedEmbedding_injectiveproof · cited by 6
- Algebra.FormallyUnramified.isReduced_of_fieldproof · cited by 6
- IsGaloisGroup.of_isFractionRingproof · cited by 5
- NumberField.InfinitePlace.mk_eq_iffproof · cited by 5