Theorems · Theorem · commutative algebra
Ideal.span_singleton_eq_top
∀ {α : Type u} [inst : CommSemiring α] {x : α}, Ideal.span {x} = ⊤ ↔ IsUnit x- Defined in
- Mathlib.RingTheory.Ideal.Span
- Cited by
- 10 results in Mathlib
- Foundations
- Depth 30 from the axioms · uses propext, Quot.sound
- Assumes
- CommSemiring
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.
- Setstatement · cited by 53,352
- CommSemiringstatement and proof · cited by 10,911
- Top.topstatement and proof · cited by 9,680
- Idealstatement and proof · cited by 4,748
- IsUnitstatement · cited by 1,602
- Ideal.spanstatement and proof · cited by 948
- eq_top_iffproof · cited by 236
- Ideal.span_singleton_oneproof · cited by 21
- isUnit_iff_dvd_oneproof · cited by 21
- Ideal.span_singleton_le_span_singletonproof · cited by 15
Cited by10
Results whose statement or proof uses this declaration.
- gcd_isUnit_iffproof · cited by 5
- IsRelPrime.isCoprimeproof · cited by 5
- PrincipalIdealRing.isMaximal_of_irreducibleproof · cited by 4
- Valuation.Uniformizer.is_generatorproof · cited by 4
- exists_max_ideal_of_mem_nonunitsproof · cited by 3
- Ideal.exists_isMaximal_dvd_of_dvd_absNormproof · cited by 2
- Ideal.FinrankQuotientMap.span_eq_topproof · cited by 1
- Algebra.exists_etale_isIdempotentElem_forall_liesOver_eq_aux₂proof · cited by 1
- Module.supportDim_add_length_eq_supportDim_of_isRegularproof · cited by 1