Theorems · Theorem · commutative algebra
Module.Flat.isSMulRegular_of_nonZeroDivisors
∀ {R : Type u_1} {M : Type u_2} [inst : CommRing R] [inst_1 : AddCommGroup M] [inst_2 : Module R M] {r : R},
r ∈ nonZeroDivisors R → ∀ [Module.Flat R M], IsSMulRegular M rScalar multiplication m ↦ r • m by a nonzerodivisor r is injective on a flat module.
- Defined in
- Mathlib.RingTheory.Flat.TorsionFree
- Cited by
- 2 results in Mathlib
- Foundations
- Depth 103 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.
- Modulestatement and proof · cited by 20,661
- CommRingstatement and proof · cited by 17,173
- AddCommGroupstatement and proof · cited by 12,871
- Submonoidstatement · cited by 3,086
- le_reflproof · cited by 2,061
- nonZeroDivisorsstatement and proof · cited by 895
- Module.Flatstatement and proof · cited by 279
- IsSMulRegularstatement · cited by 128
- le_nonZeroDivisors_iff_isRegularproof · cited by 2
- Module.Flat.isSMulRegular_of_isRegularproof · cited by 1
Cited by2
Results whose statement or proof uses this declaration.
- Module.Flat.torsion_eq_botproof · cited by 2