Theorems · Theorem · commutative algebra
Submodule.isAssociatedPrime_iff
∀ {R : Type u_1} {M : Type u_2} [inst : CommSemiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M]
{N : Submodule R M} {I : Ideal R} [IsNoetherianRing R], N.IsAssociatedPrime I ↔ I.IsPrime ∧ ∃ x, I = N.colon {x}- 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.
Cites17
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Setstatement and proof · 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 and proof · cited by 7,192
- Idealstatement and proof · cited by 4,748
- Ideal.IsPrimestatement and proof · cited by 827
- IsNoetherianRingstatement and proof · cited by 268
- Set.mem_singleton_iffproof · cited by 172
- Ideal.radicalproof · cited by 121
- Submodule.colonstatement and proof · cited by 80
- Submodule.IsAssociatedPrime.casesOnproof · cited by 8
Cited by1
Results whose statement or proof uses this declaration.
- isAssociatedPrime_iffproof · cited by 5