Theorems · Theorem · commutative algebra
exists_le_isAssociatedPrime_of_isNoetherianRing
∀ (R : Type u_1) [inst : CommSemiring R] {M : Type u_2} [inst_1 : AddCommMonoid M] [inst_2 : Module R M]
[H : IsNoetherianRing R] (x : M), x ≠ 0 → ∃ P, IsAssociatedPrime P M ∧ ⊥.colon {x} ≤ P- Cited by
- 2 results in Mathlib
- Foundations
- Depth 90 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites21
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
- Top.topproof · cited by 9,680
- Submodulestatement and proof · cited by 7,192
- Set.ofPredproof · cited by 6,101
- Idealstatement and proof · cited by 4,748
- Bot.botstatement and proof · cited by 4,720
- LE.le.transproof · cited by 3,151
- smul_zeroproof · cited by 665
- Eq.leproof · cited by 605
Cited by2
Results whose statement or proof uses this declaration.
- associatedPrimes.nonemptyproof · cited by 4
- biUnion_associatedPrimes_eq_zero_divisorsproof · cited by 2