Theorems · Theorem · commutative algebra
Ideal.span_singleton_lt_span_singleton
∀ {α : Type u} [inst : CommSemiring α] [IsDomain α] {x y : α}, Ideal.span {x} < Ideal.span {y} ↔ DvdNotUnit y x- Defined in
- Mathlib.RingTheory.Ideal.Maximal
- Cited by
- 4 results in Mathlib
- Foundations
- Depth 30 from the axioms · uses propext, Quot.sound
- Assumes
- CommSemiringIsDomain
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites9
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Setstatement · cited by 53,352
- CommSemiringstatement and proof · cited by 10,911
- Idealstatement · cited by 4,748
- IsDomainstatement and proof · cited by 2,196
- Ideal.spanstatement and proof · cited by 948
- lt_iff_le_not_geproof · cited by 35
- DvdNotUnitstatement and proof · cited by 33
- Ideal.span_singleton_le_span_singletonproof · cited by 15
- dvd_and_not_dvd_iffproof · cited by 2
Cited by4
Results whose statement or proof uses this declaration.
- IsBezout.TFAEproof · cited by 2
- WfDvdMonoid.of_setOfPred_isPrincipal_wellFoundedOn_gtproof · cited by 1
- Ideal.setOfPred_isPrincipal_wellFoundedOn_gtproof · cited by 1
- ringKrullDim_natproof · cited by 0