File: Mathlib/RingTheory/Ideal/AssociatedPrime.lean

full_name: IsAssociatedPrime
code: /-- `IsAssociatedPrime I M` if the prime ideal `I` is the annihilator of some `x : M`. -/
def IsAssociatedPrime : Prop 
goal: Prop

full_name: associatedPrimes
code: /-- The set of associated primes of a module. -/
def associatedPrimes : Set (Ideal R) 
goal: Set (Ideal R)

full_name: AssociatePrimes.mem_iff
code: theorem AssociatePrimes.mem_iff : I ∈ associatedPrimes R M ↔ IsAssociatedPrime I M 
goal: I ∈ associatedPrimes R M ↔ IsAssociatedPrime I M

full_name: IsAssociatedPrime.isPrime
code: theorem IsAssociatedPrime.isPrime (h : IsAssociatedPrime I M) : I.IsPrime 
goal: I.IsPrime

full_name: IsAssociatedPrime.map_of_injective
code: theorem IsAssociatedPrime.map_of_injective (h : IsAssociatedPrime I M) (hf : Function.Injective f) :
    IsAssociatedPrime I M' 
goal: IsAssociatedPrime I M'

full_name: LinearEquiv.isAssociatedPrime_iff
code: theorem LinearEquiv.isAssociatedPrime_iff (l : M ≃ₗ[R] M') :
    IsAssociatedPrime I M ↔ IsAssociatedPrime I M' 
goal: IsAssociatedPrime I M ↔ IsAssociatedPrime I M'

full_name: not_isAssociatedPrime_of_subsingleton
code: theorem not_isAssociatedPrime_of_subsingleton [Subsingleton M] : ¬IsAssociatedPrime I M 
goal: ¬IsAssociatedPrime I M

full_name: exists_le_isAssociatedPrime_of_isNoetherianRing
code: theorem exists_le_isAssociatedPrime_of_isNoetherianRing [H : IsNoetherianRing R] (x : M)
    (hx : x ≠ 0) : ∃ P : Ideal R, IsAssociatedPrime P M ∧ (R ∙ x).annihilator ≤ P 
goal: ∃ P, IsAssociatedPrime P M ∧ (Submodule.span R {x}).annihilator ≤ P

full_name: associatedPrimes.subset_of_injective
code: theorem associatedPrimes.subset_of_injective (hf : Function.Injective f) :
    associatedPrimes R M ⊆ associatedPrimes R M' 
goal: associatedPrimes R M ⊆ associatedPrimes R M'

full_name: LinearEquiv.AssociatedPrimes.eq
code: theorem LinearEquiv.AssociatedPrimes.eq (l : M ≃ₗ[R] M') :
    associatedPrimes R M = associatedPrimes R M' 
goal: associatedPrimes R M = associatedPrimes R M'

full_name: associatedPrimes.eq_empty_of_subsingleton
code: theorem associatedPrimes.eq_empty_of_subsingleton [Subsingleton M] : associatedPrimes R M = ∅ 
goal: associatedPrimes R M = ∅

full_name: associatedPrimes.nonempty
code: theorem associatedPrimes.nonempty [IsNoetherianRing R] [Nontrivial M] :
    (associatedPrimes R M).Nonempty 
goal: (associatedPrimes R M).Nonempty

full_name: biUnion_associatedPrimes_eq_zero_divisors
code: theorem biUnion_associatedPrimes_eq_zero_divisors [IsNoetherianRing R] :
    ⋃ p ∈ associatedPrimes R M, p = { r : R | ∃ x : M, x ≠ 0 ∧ r • x = 0 } 
goal: ⋃ p ∈ associatedPrimes R M, ↑p = {r | ∃ x, x ≠ 0 ∧ r • x = 0}

full_name: IsAssociatedPrime.annihilator_le
code: theorem IsAssociatedPrime.annihilator_le (h : IsAssociatedPrime I M) :
    (⊤ : Submodule R M).annihilator ≤ I 
goal: ⊤.annihilator ≤ I

full_name: IsAssociatedPrime.eq_radical
code: theorem IsAssociatedPrime.eq_radical (hI : I.IsPrimary) (h : IsAssociatedPrime J (R ⧸ I)) :
    J = I.radical 
goal: J = I.radical

full_name: associatedPrimes.eq_singleton_of_isPrimary
code: theorem associatedPrimes.eq_singleton_of_isPrimary [IsNoetherianRing R] (hI : I.IsPrimary) :
    associatedPrimes R (R ⧸ I) = {I.radical} 
goal: associatedPrimes R (R ⧸ I) = {I.radical}