Theorems · Theorem · commutative algebra
smul_eq_zero_iff_right
∀ {R : Type u_1} {M : Type u_3} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M] {r : R} {m : M}
[Module.IsTorsionFree R M] [IsCancelMulZero R], r ≠ 0 → (r • m = 0 ↔ m = 0)- Defined in
- Mathlib.Algebra.Module.Torsion.Free
- Cited by
- 17 results in Mathlib
- Foundations
- Depth 17 from the axioms · uses propext
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_eq_zero_iff_rightproof · cited by 2
Cited by17
Results whose statement or proof uses this declaration.
- meromorphicOrderAt_eq_int_iffproof · cited by 31
- meromorphicOrderAt_eq_top_iffproof · cited by 20
- MeromorphicAt.invproof · cited by 9
- tendsto_zero_of_isBoundedUnder_smul_of_tendsto_coboundedproof · cited by 5
- AnalyticAt.meromorphicTrailingCoeffAt_of_eq_nhdsNEproof · cited by 4
- MeromorphicAt.eventually_eq_zero_or_eventually_ne_zeroproof · cited by 3
- AnalyticAt.exists_eventuallyEq_pow_smul_nonzero_iffproof · cited by 3
- Polynomial.Chebyshev.integral_eval_T_real_measureT_of_ne_zeroproof · cited by 3
- smul_ne_zero_iff_rightproof · cited by 1
- isStarNormal_iff_commute_realPart_imaginaryPartproof · cited by 1
- LinearIndependent.pair_smul_iffproof · cited by 1
- Affine.Simplex.centroid_weighted_vsub_eq_zeroproof · cited by 1