Theorems · Theorem · commutative algebra
Ideal.IsMaximal.out
∀ {α : Type u} {inst : Semiring α} {I : Ideal α} [self : I.IsMaximal], IsCoatom IThe maximal ideal is a coatom in the ordering on ideals; that is, it is not the entire ring, and there are no other proper ideals strictly containing it.
- Defined in
- Mathlib.RingTheory.Ideal.Maximal
- Cited by
- 16 results in Mathlib
- Foundations
- Depth 25 from the axioms · uses propext, Quot.sound
- Assumes
- Ideal.IsMaximal
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites4
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
- Ideal.IsMaximalstatement and proof · cited by 452
- IsCoatomstatement · cited by 114
Cited by16
Results whose statement or proof uses this declaration.
- Ideal.IsMaximal.isPrimeproof · cited by 53
- Ideal.IsMaximal.eq_of_leproof · cited by 39
- Ideal.isMaximal_defproof · cited by 11
- Ideal.comap_isMaximal_of_surjectiveproof · cited by 8
- Ideal.map_eq_top_or_isMaximal_of_surjectiveproof · cited by 6
- Ideal.isMaximal_of_isIntegral_of_isMaximal_comapproof · cited by 5
- IsLocalRing.of_unique_max_idealproof · cited by 4
- IsLocalization.isMaximal_iff_isMaximal_disjointproof · cited by 3
- IsLocalRing.isMaximal_iffproof · cited by 3
- Ideal.exists_disjoint_powers_of_span_eq_topproof · cited by 3
- Ideal.comap_map_eq_self_of_isMaximalproof · cited by 2
- Ideal.mem_jacobson_iffproof · cited by 2