Theorems · Theorem · commutative algebra

Submodule.IsMinimalPrimaryDecomposition.mem_image_radical_colon_iff

Deprecated since 2026-01-19Use Submodule.IsMinimalPrimaryDecomposition.image_radical_eq_associated_primes instead.

∀ {R : Type u_1} {M : Type u_2} [inst : CommSemiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M]
  {N : Submodule R M} {t : Finset (Submodule R M)},
  N.IsMinimalPrimaryDecomposition t → (fun J => (J.colon Set.univ).radical) '' ↑t = N.associatedPrimes

Alias of Submodule.IsMinimalPrimaryDecomposition.image_radical_eq_associated_primes. The first uniqueness theorem for primary decomposition, Theorem 4.5 in Atiyah-Macdonald: In any minimal primary decomposition I = ⨅ i, q_i, the ideals radical (q_i.colon M) are exactly the associated primes of I.

Defined in

Mathlib.RingTheory.Lasker

Cited by

0 results in Mathlib

Foundations

Depth 86 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommSemiringAddCommMonoidModule