Loogle!

Result

Found 11 declarations mentioning Algebra.IsIntegral and Ideal.IsMaximal.

• Ideal.IsMaximal.under 📋 Mathlib.RingTheory.Ideal.GoingUp
  (A : Type u_1) [CommRing A] {B : Type u_2} [CommRing B] [Algebra A B] [Algebra.IsIntegral A B] (P : Ideal B) [P.IsMaximal] : (Ideal.under A P).IsMaximal
• Ideal.IsMaximal.of_isMaximal_liesOver 📋 Mathlib.RingTheory.Ideal.GoingUp
  {A : Type u_1} [CommRing A] {B : Type u_2} [CommRing B] [Algebra A B] [Algebra.IsIntegral A B] (P : Ideal B) (p : Ideal A) [P.LiesOver p] [P.IsMaximal] : p.IsMaximal
• Ideal.IsMaximal.of_liesOver_isMaximal 📋 Mathlib.RingTheory.Ideal.GoingUp
  {A : Type u_1} [CommRing A] {B : Type u_2} [CommRing B] [Algebra A B] [Algebra.IsIntegral A B] (P : Ideal B) (p : Ideal A) [P.LiesOver p] [hpm : p.IsMaximal] [P.IsPrime] : P.IsMaximal
• Ideal.exists_ideal_liesOver_maximal_of_isIntegral 📋 Mathlib.RingTheory.Ideal.GoingUp
  {A : Type u_1} [CommRing A] (p : Ideal A) [p.IsMaximal] (B : Type u_3) [CommRing B] [Nontrivial B] [Algebra A B] [NoZeroSMulDivisors A B] [Algebra.IsIntegral A B] : ∃ P, P.IsMaximal ∧ P.LiesOver p
• Ideal.primesOver.isMaximal 📋 Mathlib.RingTheory.Ideal.GoingUp
  {A : Type u_1} [CommRing A] {p : Ideal A} [p.IsMaximal] {B : Type u_2} [CommRing B] [Algebra A B] [Algebra.IsIntegral A B] (Q : ↑(p.primesOver B)) : (↑Q).IsMaximal
• Ideal.isMaximal_comap_of_isIntegral_of_isMaximal 📋 Mathlib.RingTheory.Ideal.GoingUp
  {R : Type u_1} [CommRing R] {S : Type u_2} [CommRing S] [Algebra R S] [Algebra.IsIntegral R S] (I : Ideal S) [hI : I.IsMaximal] : (Ideal.comap (algebraMap R S) I).IsMaximal
• Ideal.isMaximal_of_isIntegral_of_isMaximal_comap 📋 Mathlib.RingTheory.Ideal.GoingUp
  {R : Type u_1} [CommRing R] {S : Type u_2} [CommRing S] [Algebra R S] [Algebra.IsIntegral R S] (I : Ideal S) [I.IsPrime] (hI : (Ideal.comap (algebraMap R S) I).IsMaximal) : I.IsMaximal
• Ideal.exists_ideal_over_maximal_of_isIntegral 📋 Mathlib.RingTheory.Ideal.GoingUp
  {R : Type u_1} [CommRing R] {S : Type u_2} [CommRing S] [Algebra R S] [Algebra.IsIntegral R S] (P : Ideal R) [P_max : P.IsMaximal] (hP : RingHom.ker (algebraMap R S) ≤ P) : ∃ Q, Q.IsMaximal ∧ Ideal.comap (algebraMap R S) Q = P
• primesOver_finite 📋 Mathlib.RingTheory.DedekindDomain.Ideal
  {A : Type u_4} [CommRing A] (p : Ideal A) [hpm : p.IsMaximal] (B : Type u_5) [CommRing B] [IsDedekindDomain B] [Algebra A B] [NoZeroSMulDivisors A B] [Algebra.IsIntegral A B] : (p.primesOver B).Finite
• primesOver_ncard_ne_zero 📋 Mathlib.RingTheory.DedekindDomain.Ideal
  {A : Type u_4} [CommRing A] (p : Ideal A) [hpm : p.IsMaximal] (B : Type u_5) [CommRing B] [IsDedekindDomain B] [Algebra A B] [NoZeroSMulDivisors A B] [Algebra.IsIntegral A B] : (p.primesOver B).ncard ≠ 0
• one_le_primesOver_ncard 📋 Mathlib.RingTheory.DedekindDomain.Ideal
  {A : Type u_4} [CommRing A] (p : Ideal A) [hpm : p.IsMaximal] (B : Type u_5) [CommRing B] [IsDedekindDomain B] [Algebra A B] [NoZeroSMulDivisors A B] [Algebra.IsIntegral A B] : 1 ≤ (p.primesOver B).ncard

About

Loogle searches Lean and Mathlib definitions and theorems.

You can use Loogle from within the Lean4 VSCode language extension using (by default) Ctrl-K Ctrl-S. You can also try the #loogle command from LeanSearchClient, the CLI version, the Loogle VS Code extension, the lean.nvim integration or the Zulip bot.

Usage

Loogle finds definitions and lemmas in various ways:

1. By constant:
   🔍 Real.sin
   finds all lemmas whose statement somehow mentions the sine function.

2. By lemma name substring:
   🔍 "differ"
   finds all lemmas that have "differ" somewhere in their lemma name.

3. By subexpression:
   🔍 _ * (_ ^ _)
   finds all lemmas whose statements somewhere include a product where the second argument is raised to some power.

   The pattern can also be non-linear, as in
   🔍 Real.sqrt ?a * Real.sqrt ?a

   If the pattern has parameters, they are matched in any order. Both of these will find List.map:
   🔍 (?a -> ?b) -> List ?a -> List ?b
   🔍 List ?a -> (?a -> ?b) -> List ?b

4. By main conclusion:
   🔍 |- tsum _ = _ * tsum _
   finds all lemmas where the conclusion (the subexpression to the right of all → and ∀) has the given shape.

   As before, if the pattern has parameters, they are matched against the hypotheses of the lemma in any order; for example,
   🔍 |- _ < _ → tsum _ < tsum _
   will find tsum_lt_tsum even though the hypothesis f i < g i is not the last.

If you pass more than one such search filter, separated by commas Loogle will return lemmas which match all of them. The search
🔍 Real.sin, "two", tsum, _ * _, _ ^ _, |- _ < _ → _
would find all lemmas which mention the constants Real.sin and tsum, have "two" as a substring of the lemma name, include a product and a power somewhere in the type, and have a hypothesis of the form _ < _ (if there were any such lemmas). Metavariables (?a) are assigned independently in each filter.

The #lucky button will directly send you to the documentation of the first hit.

Source code

You can find the source code for this service at https://github.com/nomeata/loogle. The https://loogle.lean-lang.org/ service is provided by the Lean FRO.

This is Loogle revision 19971e9 serving mathlib revision 1bd7adb