Theorems · Theorem · ring theory
faithfulSMul_iff_algebraMap_injective
∀ (R : Type u_1) (A : Type u_2) [inst : CommSemiring R] [inst_1 : Semiring A] [inst_2 : Algebra R A], FaithfulSMul R A ↔ Function.Injective ⇑(algebraMap R A)
- Defined in
- Mathlib.Algebra.Algebra.Basic
- Cited by
- 17 results in Mathlib
- Foundations
- Depth 20 from the axioms · uses propext, Quot.sound
- Assumes
- CommSemiringSemiringAlgebra
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites9
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
- Semiringstatement and proof · cited by 13,802
- 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
- FaithfulSMulstatement · cited by 340
- faithfulSMul_iff_injective_smul_oneproof · cited by 8
- Algebra.algebraMap_eq_smul_one'proof · cited by 3
Cited by17
Results whose statement or proof uses this declaration.
- FaithfulSMul.algebraMap_injectiveproof · cited by 198
- IsTranscendenceBasis.lift_cardinalMk_eq_trdegproof · cited by 7
- IsLocalization.rank_eqproof · cited by 7
- Module.isTorsionFree_iff_algebraMap_injectiveproof · cited by 6
- isField_of_isIntegral_of_isFieldproof · cited by 4
- Module.supportDim_self_eq_ringKrullDimproof · cited by 3
- isFractionRing_of_exists_eq_algebraMap_or_inv_eq_algebraMap_of_injectiveproof · cited by 2
- AlgebraicIndependent.isTranscendenceBasis_of_lift_trdeg_leproof · cited by 2
- Algebra.isUnramifiedAt_botproof · cited by 2
- RingHom.IsIntegral.comap_surjectiveproof · cited by 1
- PrimeSpectrum.comap_surjective_iff_injective_of_finiteproof · cited by 1
- FaithfulSMul.of_field_isFractionRingproof · cited by 1