Theorems · Theorem · commutative algebra
Module.associatedPrimes.mem_associatedPrimes_atPrime_of_mem_associatedPrimes
∀ {R : Type u_1} [inst : CommRing R] {M : Type u_3} [inst_1 : AddCommGroup M] [inst_2 : Module R M] {p : Ideal R}
[inst_3 : p.IsPrime],
p ∈ associatedPrimes R M →
IsLocalRing.maximalIdeal (Localization.AtPrime p) ∈
associatedPrimes (Localization.AtPrime p) (LocalizedModule.AtPrime p M)- Cited by
- 1 results in Mathlib
- Foundations
- Depth 87 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites14
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Setstatement · cited by 53,352
- Modulestatement and proof · cited by 20,661
- CommRingstatement and proof · cited by 17,173
- AddCommGroupstatement and proof · cited by 12,871
- Idealstatement and proof · cited by 4,748
- Ideal.IsPrimestatement and proof · cited by 827
- Ideal.primeComplstatement and proof · cited by 462
- Localization.AtPrimestatement and proof · cited by 299
- IsLocalRing.maximalIdealstatement and proof · cited by 297
- LocalizedModule.mkLinearMapproof · cited by 76
- associatedPrimesstatement and proof · cited by 21
- Localization.AtPrime.under_maximalIdealproof · cited by 9
Cited by1
Results whose statement or proof uses this declaration.
- IsSMulRegular.subsingleton_linearMap_iffproof · cited by 2