Theorems · Theorem · commutative algebra

monotone_stabilizes_iff_noetherian

∀ {R : Type u_1} {M : Type u_2} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M],
  (∀ (f : ℕ →o Submodule R M), ∃ n, ∀ (m : ℕ), n ≤ m → f n = f m) ↔ IsNoetherian R M

A module is Noetherian iff every increasing chain of submodules stabilizes.

Defined in

Mathlib.RingTheory.Noetherian.Defs

Cited by

8 results in Mathlib

Foundations

Depth 82 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

SemiringAddCommMonoidModule