isNoetherian_def — Mathlib

English

An R-submodule N of M is Noetherian iff every submodule s of M with s ≤ N is FG (finitely generated).

Русский

Подмодуль N над R модуля M Ноетерен тод, если каждый подмодуль s ⊆ M с s ≤ N образует FG.

LaTeX

$$IsNoetherian(R,N) ↔ ∀ s ⊆ M, s ≤ N → s.FG$$

Lean4

/-- An R-module is Noetherian iff all its submodules are finitely-generated. -/
theorem isNoetherian_def : IsNoetherian R M ↔ ∀ s : Submodule R M, s.FG :=
  ⟨fun h => h.noetherian, IsNoetherian.mk⟩

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