Theorems · Theorem · commutative algebra
Ideal.exists_maximal
∀ (α : Type u) [inst : Semiring α] [Nontrivial α], ∃ M, M.IsMaximal
Krull's theorem: a nontrivial ring has a maximal ideal.
- Defined in
- Mathlib.RingTheory.Ideal.Maximal
- Cited by
- 6 results in Mathlib
- Foundations
- Depth 75 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- SemiringNontrivial
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites7
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
- Idealstatement and proof · cited by 4,748
- Bot.botproof · cited by 4,720
- Nontrivialstatement and proof · cited by 2,416
- Ideal.IsMaximalstatement and proof · cited by 452
- Ideal.exists_le_maximalproof · cited by 47
- bot_ne_topproof · cited by 24
Cited by6
Results whose statement or proof uses this declaration.
- Polynomial.separable_mapproof · cited by 6
- PrimeSpectrum.nonempty_iff_nontrivialproof · cited by 3
- Ideal.exists_ideal_over_prime_of_isIntegral_of_isDomainproof · cited by 2
- PrimeSpectrum.nontrivial_iff_mem_rangeComapproof · cited by 1
- finite_of_algHom_finiteType_of_isJacobsonRingproof · cited by 1
- IntermediateField.LinearDisjoint.isField_of_forallproof · cited by 0