Theorems · Theorem · commutative algebra
CharZero.of_isAddTorsionFree
∀ (R : Type u_1) (M : Type u_3) [inst : Semiring R] [inst_1 : AddCommGroup M] [Module R M] [Nontrivial M] [IsAddTorsionFree M], CharZero R
Only a ring of characteristic zero can have a non-trivial module without additive or scalar torsion.
- Defined in
- Mathlib.Algebra.Module.Torsion.Free
- Cited by
- 2 results in Mathlib
- Foundations
- Depth 41 from the axioms · uses propext, Classical.choice, 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.
- Modulestatement and proof · cited by 20,661
- Semiringstatement and proof · cited by 13,802
- AddCommGroupstatement and proof · cited by 12,871
- AddCommMonoidproof · cited by 12,281
- Nontrivialstatement and proof · cited by 2,416
- CharZerostatement · cited by 932
- IsAddTorsionFreestatement and proof · cited by 155
- natCast_zsmulproof · cited by 118
- Nat.cast_smul_eq_nsmulproof · cited by 110
- exists_neproof · cited by 101
- smul_left_injectiveproof · cited by 25
Cited by2
Results whose statement or proof uses this declaration.
- Ideal.exists_isMaximal_dvd_of_dvd_absNormproof · cited by 2
- Ideal.exists_prime_and_absNorm_eq_powproof · cited by 1