Theorems · Theorem · ring theory
Module.isTorsionFree_iff_algebraMap_injective
∀ {R : Type u_1} {A : Type u_2} [inst : CommRing R] [inst_1 : Ring A] [inst_2 : Algebra R A] [IsDomain R] [IsDomain A],
Module.IsTorsionFree R A ↔ Function.Injective ⇑(algebraMap R A)- Defined in
- Mathlib.Algebra.Algebra.Basic
- Cited by
- 6 results in Mathlib
- Foundations
- Depth 27 from the axioms · uses propext, Classical.choice, 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 and proof · cited by 62,936
- CommRingstatement and proof · cited by 17,173
- Algebrastatement and proof · cited by 11,388
- RingHomstatement · cited by 10,189
- Ringstatement and proof · cited by 7,463
- Algebra.algebraMapstatement and proof · cited by 4,706
- IsDomainstatement and proof · cited by 2,196
- Module.IsTorsionFreestatement · cited by 600
- faithfulSMul_iff_algebraMap_injectiveproof · cited by 17
- Module.isTorsionFree_iff_faithfulSMulproof · cited by 2
Cited by6
Results whose statement or proof uses this declaration.
- Algebra.trace_quotient_eq_of_isDedekindDomainproof · cited by 3
- AdjoinRoot.noZeroSMulDivisors_of_prime_of_degree_ne_zeroproof · cited by 1
- Algebra.dvd_algebraMap_intNorm_selfproof · cited by 1
- groupCohomology.exists_mul_galRestrict_of_norm_eq_oneproof · cited by 0
- NoZeroSMulDivisors.iff_algebraMap_injectiveproof · cited by 0
- Module.IsTorsionFree.of_isLocalizationproof · cited by 0