Remark

Let A be a ring. An Artinian left A-module M with finitely generated countermodule has finite length: indeed, the ring of homotheties $A_M$ of M is left Artinian (Proposition 6), and M is an Artinian module over $A_M$; by VIII, p. 7, Proposition 4, the module M has finite length over $A_M$ and therefore also over A.

In particular, every finitely generated Artinian module over a commutative ring has finite length. By contrast, a finitely generated Artinian module over a noncommutative ring is not necessarily of finite length (VIII, p. 16, Exercise 12).