Theorems · Theorem · commutative algebra
Ideal.ne_top_iff_one
∀ {α : Type u} [inst : Semiring α] (I : Ideal α), I ≠ ⊤ ↔ 1 ∉ I- Defined in
- Mathlib.RingTheory.Ideal.Lattice
- Cited by
- 16 results in Mathlib
- Foundations
- Depth 27 from the axioms · uses propext, Quot.sound
- Assumes
- Semiring
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites4
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Semiringstatement and proof · cited by 13,802
- Top.topstatement · cited by 9,680
- Idealstatement and proof · cited by 4,748
- Ideal.eq_top_iff_oneproof · cited by 56
Cited by16
Results whose statement or proof uses this declaration.
- Ideal.comap_ne_topproof · cited by 7
- Module.eq_of_localization_maximalproof · cited by 4
- RingHom.ker_ne_topproof · cited by 4
- Ideal.span_singleton_ne_topproof · cited by 3
- MaximalSpectrum.iInf_localization_eq_botproof · cited by 3
- MaximalSpectrum.toPiLocalization_injectiveproof · cited by 3
- Ideal.eq_bot_or_topproof · cited by 2
- PrimeSpectrum.sigmaToPi_injectiveproof · cited by 2
- PrimeSpectrum.exists_maximal_notMem_range_sigmaToPi_of_infiniteproof · cited by 2
- Polynomial.IsDistinguishedAt.isWeierstrassDivisorAtproof · cited by 1
- Ideal.IsMaximal.comap_piEvalRingHomproof · cited by 1
- AlgebraicClosure.spanCoeffs_ne_topproof · cited by 1