Theorems · Theorem · commutative algebra
isAssociatedPrime_iff
∀ {R : Type u_1} [inst : CommSemiring R] {I : Ideal R} {M : Type u_2} [inst_1 : AddCommMonoid M] [inst_2 : Module R M]
[IsNoetherianRing R], IsAssociatedPrime I M ↔ I.IsPrime ∧ ∃ x, I = ⊥.colon {x}- Cited by
- 5 results in Mathlib
- Foundations
- Depth 88 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites12
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
- AddCommMonoidstatement and proof · cited by 12,281
- CommSemiringstatement and proof · cited by 10,911
- Submodulestatement · cited by 7,192
- Idealstatement and proof · cited by 4,748
- Bot.botstatement · cited by 4,720
- Ideal.IsPrimestatement · cited by 827
- IsNoetherianRingstatement and proof · cited by 268
- Submodule.colonstatement · cited by 80
- IsAssociatedPrimestatement · cited by 12
- Submodule.isAssociatedPrime_iffproof · cited by 1
Cited by5
Results whose statement or proof uses this declaration.
- biUnion_associatedPrimes_eq_zero_divisorsproof · cited by 2
- IsSMulRegular.subsingleton_linearMap_iffproof · cited by 2
- IsNoetherianRing.exists_relSeries_isQuotientEquivQuotientPrimeproof · cited by 1
- Ideal.bot_lt_annihilator_of_disjoint_nonZeroDivisorsproof · cited by 1
- isAssociatedPrime_iff_exists_injective_linearMapproof · cited by 0