Theorems · Theorem · ring theory

IsNoetherian.iff_rank_lt_aleph0

∀ {K : Type u} {V : Type v} [inst : DivisionRing K] [inst_1 : AddCommGroup V] [inst_2 : Module K V],
  IsNoetherian K V ↔ Module.rank K V < Cardinal.aleph0

A module over a division ring is Noetherian if and only if its dimension (as a cardinal) is strictly less than the first infinite cardinal ℵ₀.

Defined in

Mathlib.FieldTheory.Finiteness

Cited by

2 results in Mathlib

Foundations

Depth 111 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

DivisionRingAddCommGroupModule