Theorems · Theorem · commutative algebra
smul_right_injective
∀ {R : Type u_1} (M : Type u_3) [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M] {r : R}
[Module.IsTorsionFree R M] [IsCancelMulZero R], r ≠ 0 → Function.Injective fun x => r • x- Defined in
- Mathlib.Algebra.Module.Torsion.Free
- Cited by
- 24 results in Mathlib
- Foundations
- Depth 15 from the axioms · uses no axioms
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites7
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
- AddCommMonoidstatement and proof · cited by 12,281
- Module.IsTorsionFreestatement and proof · cited by 600
- IsCancelMulZerostatement and proof · cited by 177
- IsRegular.of_ne_zeroproof · cited by 15
- IsRegular.smul_right_injectiveproof · cited by 2
Cited by24
Results whose statement or proof uses this declaration.
- IsAddTorsionFree.of_isTorsionFreeproof · cited by 22
- Submodule.reflection_eq_self_iffproof · cited by 3
- LinearMap.lsmul_injectiveproof · cited by 3
- RootPairing.pairing_smul_root_eq_of_not_linearIndependentproof · cited by 3
- Module.eq_of_mapsTo_reflection_of_memproof · cited by 2
- left_mem_openSegment_iffproof · cited by 2
- RootPairing.finrank_range_polarization_eq_finrank_span_corootproof · cited by 2
- RootPairing.pairing_one_four_iff'proof · cited by 2
- RootPairing.pairing_two_two_iffproof · cited by 2
- RootPairing.coroot_eq_polarizationEquiv_apply_rootproof · cited by 2
- Convex.second_derivative_within_at_symmetricproof · cited by 1
- Submodule.reflection_subproof · cited by 1