Mathlib Map

Theorems · Definition · commutative algebra

IsAssociatedPrime

{R : Type u_1} → [inst : CommSemiring R] → Ideal R → (M : Type u_2) → [inst_1 : AddCommMonoid M] → [Module R M] → Prop

IsAssociatedPrime I M if the prime ideal I is the radical of the annihilator of some x : M.

Defined in
Mathlib.RingTheory.Ideal.AssociatedPrime.Basic
Cited by
12 results in Mathlib
Foundations
Depth 16 from the axioms · uses propext, Quot.sound
Assumes
CommSemiringAddCommMonoidModule
Mathlib docsSourceLoogleLeanSearch

Around this declaration

Dashed lines are statement dependencies; solid lines are citations in proofs.

associatedPrimes · cited by 21associatedPrimesisAssociatedPrime_iff · cited by 5isAssociatedPrime_iffassociatedPrimes.nonempty · cited by 4associatedPrimes.nonemptyIsAssociatedPrime.isPrime · cited by 3IsAssociatedPrime.isPrimeAssociatedPrimes.mem_iff · cited by 3AssociatedPrimes.mem_iffbiUnion_associatedPrimes_eq_zero_divisors · cited by 2biUnion_associatedPrimes_…IsAssociatedPrime.map_of_injective · cited by 2IsAssociatedPrime.map_of_…exists_le_isAssociatedPrime_of_isNoetherianRing · cited by 2exists_le_isAssociatedPri…IsAssociatedPrime.annihilator_le · cited by 1IsAssociatedPrime.annihil…IsAssociatedPrime.eq_radical · cited by 1IsAssociatedPrime.eq_radi…not_isAssociatedPrime_of_subsingleton · cited by 1not_isAssociatedPrime_of_…LinearEquiv.isAssociatedPrime_iff · cited by 0LinearEquiv.isAssociatedP…isAssociatedPrime_iff_exists_injective_linearMap · cited by 0isAssociatedPrime_iff_exi…Module · cited by 20661ModuleAddCommMonoid · cited by 12281AddCommMonoidCommSemiring · cited by 10911CommSemiringIdeal · cited by 4748IdealBot.bot · cited by 4720Bot.botSubmodule.IsAssociatedPrime · cited by 5Submodule.IsAssociatedPri…IsAssociatedPrimeCITED BYCITES

Cites6

Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.

Cited by13

Results whose statement or proof uses this declaration.