Theorems · Theorem · commutative algebra

associatedPrimes.subset_union_of_exact

∀ {R : Type u_1} [inst : CommSemiring R] {M : Type u_2} [inst_1 : AddCommMonoid M] [inst_2 : Module R M] {M' : Type u_3}
  [inst_3 : AddCommMonoid M'] [inst_4 : Module R M'] {f : M →ₗ[R] M'} {M'' : Type u_4} [inst_5 : AddCommMonoid M'']
  [inst_6 : Module R M''] {g : M' →ₗ[R] M''},
  Function.Injective ⇑f → Function.Exact ⇑f ⇑g → associatedPrimes R M' ⊆ associatedPrimes R M ∪ associatedPrimes R M''

If 0 → M → M' → M'' is an exact sequence, then the set of associated primes of M' is contained in the union of those of M and M''.

Defined in

Mathlib.RingTheory.Ideal.AssociatedPrime.Basic

Cited by

2 results in Mathlib

Foundations

Depth 80 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommSemiringAddCommMonoidModuleAddCommMonoidModuleAddCommMonoidModule