Theorems · Theorem · commutative algebra

Submodule.exists_eq_colon_of_mem_minimalPrimes

∀ {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} {x : M} [IsNoetherianRing R],
  I ∈ (N.colon {x}).minimalPrimes → ∃ x', I = N.colon {x'}

A minimal prime over an ideal of the form N.colon {x} in a Noetherian ring is itself an ideal of the form N.colon {x'}.

Defined in

Mathlib.RingTheory.Ideal.MinimalPrime.Colon

Cited by

1 results in Mathlib

Foundations

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

Assumes

CommSemiringAddCommMonoidModuleIsNoetherianRing