Theorems · Theorem · commutative algebra
Ideal.exists_le_maximal
∀ {α : Type u} [inst : Semiring α] (I : Ideal α), I ≠ ⊤ → ∃ M, M.IsMaximal ∧ I ≤ MKrull's theorem: if I is an ideal that is not the whole ring, then it is included in some
maximal ideal.
- Defined in
- Mathlib.RingTheory.Ideal.Maximal
- Cited by
- 47 results in Mathlib
- Foundations
- Depth 74 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
- IsCoatomproof · cited by 114
- IsCoatomic.eq_top_or_exists_le_coatomproof · cited by 9
Cited by47
Results whose statement or proof uses this declaration.
- IsLocalRing.le_maximalIdealproof · cited by 20
- PrimeSpectrum.zeroLocus_empty_iff_eq_topproof · cited by 8
- PrimeSpectrum.isClosed_singleton_iff_isMaximalproof · cited by 8
- Ring.krullDimLE_zero_and_isLocalRing_tfaeproof · cited by 7
- Ideal.nonempty_minimalPrimesproof · cited by 7
- Ideal.exists_maximalproof · cited by 6
- Module.eq_of_localization_maximalproof · cited by 4
- Submodule.mem_of_localization_maximalproof · cited by 4
- Ideal.exists_disjoint_powers_of_span_eq_topproof · cited by 3
- Ideal.maximal_of_no_maximalproof · cited by 3
- MaximalSpectrum.iInf_localization_eq_botproof · cited by 3
- exists_max_ideal_of_mem_nonunitsproof · cited by 3