Theorems · Theorem · commutative algebra
Ring.ne_bot_of_isMaximal_of_not_isField
∀ {R : Type u_5} [inst : CommSemiring R] [Nontrivial R] {M : Ideal R}, M.IsMaximal → ¬IsField R → M ≠ ⊥When a ring is not a field, the maximal ideals are nontrivial.
- Defined in
- Mathlib.RingTheory.Ideal.Basic
- Cited by
- 8 results in Mathlib
- Foundations
- Depth 79 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- CommSemiringNontrivial
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites11
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- CommSemiringstatement and proof · cited by 10,911
- Top.topproof · cited by 9,680
- Idealstatement and proof · cited by 4,748
- Bot.botstatement and proof · cited by 4,720
- Nontrivialstatement and proof · cited by 2,416
- LT.lt.neproof · cited by 872
- Ideal.IsMaximalstatement and proof · cited by 452
- bot_leproof · cited by 306
- IsFieldstatement and proof · cited by 103
- Ideal.IsMaximal.eq_of_leproof · cited by 39
- Ring.not_isField_iff_exists_ideal_bot_lt_and_lt_topproof · cited by 3
Cited by8
Results whose statement or proof uses this declaration.
- IsLocalRing.isField_iff_maximalIdeal_eqproof · cited by 8
- Ideal.bot_lt_of_maximalproof · cited by 3
- PrimeSpectrum.t1Space_iff_isFieldproof · cited by 2
- IsIntegrallyClosed.of_localization_maximalproof · cited by 2
- Ideal.IsMaximal.ne_bot_of_isIntegral_intproof · cited by 2
- exists_maximalIdeal_pow_eq_of_principalproof · cited by 1
- IntermediateField.LinearDisjoint.isField_of_forallproof · cited by 0
- Valuation.isUniformizer_of_maximalIdeal_eq_spanproof · cited by 0