Theorems · Theorem · commutative algebra
Ideal.maximal_of_no_maximal
∀ {α : Type u} [inst : Semiring α] {P : Ideal α},
(∀ (m : Ideal α), P < m → ¬m.IsMaximal) → ∀ (J : Ideal α), P < J → J = ⊤If P is not properly contained in any maximal ideal then it is not properly contained in any proper ideal
- Defined in
- Mathlib.RingTheory.Ideal.Maximal
- Cited by
- 3 results in Mathlib
- Foundations
- Depth 75 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- Semiring
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites6
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 and proof · cited by 9,680
- Idealstatement and proof · cited by 4,748
- Ideal.IsMaximalstatement and proof · cited by 452
- lt_of_lt_of_leproof · cited by 438
- Ideal.exists_le_maximalproof · cited by 47
Cited by3
Results whose statement or proof uses this declaration.
- Ring.exists_maximal_of_not_isFieldproof · cited by 2
- Ring.DimensionLEOne.localizationproof · cited by 1
- IsLocalRing.of_unique_nonzero_primeproof · cited by 1