Theorems · Theorem · commutative algebra
biUnion_associatedPrimes_eq_zero_divisors
∀ (R : Type u_1) [inst : CommSemiring R] (M : Type u_2) [inst_1 : AddCommMonoid M] [inst_2 : Module R M]
[IsNoetherianRing R], ⋃ p ∈ associatedPrimes R M, ↑p = {r | ∃ x, x ≠ 0 ∧ r • x = 0}- Cited by
- 2 results in Mathlib
- Foundations
- Depth 91 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites22
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
- SetLike.coestatement and proof · cited by 8,199
- Set.ofPredstatement and proof · cited by 6,101
- Idealstatement and proof · cited by 4,748
- Bot.botproof · cited by 4,720
- Set.iUnionstatement and proof · cited by 2,483
- Ideal.IsPrimeproof · cited by 827
- Set.iUnion_congr_Propproof · cited by 374
- IsNoetherianRingstatement and proof · cited by 268
Cited by2
Results whose statement or proof uses this declaration.
- biUnion_associatedPrimes_eq_compl_nonZeroDivisorsproof · cited by 2
- biUnion_associatedPrimes_eq_compl_regularproof · cited by 1