Theorems · Theorem · group theory
Submonoid.LocalizationMap.isCancelMulZero
∀ {M : Type u_1} [inst : CommMonoidWithZero M] {S : Submonoid M} {N : Type u_2} [inst_1 : CommMonoidWithZero N]
(f : S.LocalizationMap N) [IsCancelMulZero M], IsCancelMulZero NGiven a Localization map f : M →*₀ N for a Submonoid S ⊆ M,
if M is a cancellative monoid with zero, and all elements of S are
regular, then N is a cancellative monoid with zero.
- Cited by
- 2 results in Mathlib
- Foundations
- Depth 24 from the axioms · uses propext, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites18
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- DFunLike.coeproof · cited by 62,936
- Submonoidstatement and proof · cited by 3,086
- map_zeroproof · cited by 1,614
- CommMonoidWithZerostatement and proof · cited by 913
- IsCancelMulZerostatement and proof · cited by 177
- Submonoid.LocalizationMapstatement and proof · cited by 147
- Commute.allproof · cited by 119
- IsRightRegularproof · cited by 93
- CommMagmaproof · cited by 57
- Submonoid.LocalizationMap.map_unitsproof · cited by 46
- IsRegular.rightproof · cited by 19
- Submonoid.LocalizationMap.surjproof · cited by 15
Cited by2
Results whose statement or proof uses this declaration.
- IsLocalization.isDomain_of_le_nonZeroDivisorsproof · cited by 11
- Submonoid.LocalizationMap.uniqueFactorizationMonoidproof · cited by 1