Theorems · Theorem · commutative algebra
Ideal.IsMaximal.ne_top
∀ {α : Type u} [inst : Semiring α] {I : Ideal α}, I.IsMaximal → I ≠ ⊤- Defined in
- Mathlib.RingTheory.Ideal.Maximal
- Cited by
- 42 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.
Cites5
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.IsMaximalstatement and proof · cited by 452
- Ideal.isMaximal_defproof · cited by 11
Cited by42
Results whose statement or proof uses this declaration.
- Ideal.IsMaximal.coprime_of_neproof · cited by 5
- Valuation.Uniformizer.is_generatorproof · cited by 4
- PowerSeries.IsWeierstrassFactorization.map_ne_zeroproof · cited by 3
- IsLocalization.isMaximal_iff_isMaximal_disjointproof · cited by 3
- Ideal.IsMaximal.lt_topproof · cited by 3
- MaximalSpectrum.iInf_localization_eq_botproof · cited by 3
- Ring.not_isField_iff_exists_ideal_bot_lt_and_lt_topproof · cited by 3
- IsLocalization.AtPrime.not_isFieldproof · cited by 2
- FractionalIdeal.not_inv_le_one_of_ne_botproof · cited by 2
- Ideal.exists_isMaximal_dvd_of_dvd_absNormproof · cited by 2
- PrimeSpectrum.exists_multiset_prod_cons_le_and_prod_not_leproof · cited by 2
- Ideal.IsMaximal.of_isLocalization_of_disjointproof · cited by 2