Theorems · Inductive type · commutative algebra
NoZeroSMulDivisors
(R : Type u_4) → (M : Type u_5) → [Zero R] → [Zero M] → [SMul R M] → Prop
NoZeroSMulDivisors R M states that a scalar multiple is 0 only if either argument is 0.
This is a version of saying that M is torsion free, without assuming R is zero-divisor free.
The main application of NoZeroSMulDivisors R M, when M is a module,
is the result smul_eq_zero: a scalar multiple is 0 iff either argument is 0.
It is a generalization of the NoZeroDivisors class to heterogeneous multiplication.
- Defined in
- Mathlib.Algebra.NoZeroSMulDivisors.Defs
- Cited by
- 9 results in Mathlib
- Foundations
- Depth 1 from the axioms · uses no axioms
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites0
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
Nothing in Mathlib beyond the foundations.
Cited by11
Results whose statement or proof uses this declaration.
- NoZeroSMulDivisors.eq_zero_or_eq_zero_of_smul_eq_zerostatement and proof · cited by 2
- NoZeroSMulDivisors.casesOnstatement and proof · cited by 1
- LieAlgebra.Basis.A_diag_eq_twoproof · cited by 1
- Function.Injective.noZeroSMulDivisorsstatement and proof · cited by 0
- Ideal.inertiaDeg_le_finrankstatement and proof · cited by 0
- NoZeroSMulDivisors.recOnstatement and proof · cited by 0
- Ideal.card_primesOverFinset_le_finrankstatement and proof · cited by 0
- Ideal.ramificationIdx_le_finrankstatement and proof · cited by 0
- noZeroSMulDivisors_iffstatement and proof · cited by 0
- noZeroSMulDivisors_iff_right_eq_zero_of_smulstatement · cited by 0
- Equiv.noZeroSMulDivisorsstatement and proof · cited by 0