Proposition 4

a) Let A be a left Noetherian ring and M a finitely generated left A-module. The module M is Noetherian, and every submodule of M is finitely generated.

b) Let A be a left Artinian ring and M a left A-module. The following properties are equivalent: the module M is finitely generated; the module M is Artinian; the module M has finite length; the module M is Noetherian.

Let us prove a). Every monogenous submodule of M is isomorphic to a quotient of $A_s$, hence is Noetherian by Proposition 3 of VIII, p. 3. The module M is a finite sum of such submodules; it is therefore Noetherian by the corollary (VIII, p. 4) of Proposition 3. Every submodule of M is then finitely generated (VIII, p. 3, Proposition 2)

Now, suppose that the ring A is left Artinian. We see, as in the previous section, that if the A-module M is finitely generated, then it is Artinian. If it is Artinian, then it has finite length: indeed, its monogenous submodules are isomorphic to quotients of $A_s$ and are therefore of finite length less than that of $A_s$, and the assertion follows from Lemma 3. Every module of finite length is Noetherian, and every Noetherian module is finitely generated. This proves b).