# Chain of equivalences in 12.B of Matsumura's "Commutative Algebra"

I'm reading Matsumura's "Commutative Algebra", specifically the chapter on Dimension. However I am having some trouble in the section 12.B, where it is presented the chain of equivalences for a module $$M$$ over a Noetherian (commutative) ring $$A$$:

1. $$M$$ is an $$A$$-module of finite length;

2. $$A/\operatorname{Ann}(M)$$ is an Artinian ring;

3. $$\operatorname{dim}(M) = 0$$.

What I am not being able to justify is the implication (2) $$\implies$$ (1). I know that $$M$$ has finite length iff it is noetherian and artinian and also, since $$A$$ is noetherian, so is $$M$$. By definition of $$\operatorname{Ann}(M)$$, the submodules of $$M$$ seen as $$A$$-module are the same as the submodules of $$M$$ seen as an $$A/ \operatorname{Ann}(M)$$-module, and so $$M$$ is an artinian $$A$$-module iff it is an artinian $$A/ \operatorname{Ann}(M)$$-module.

I don't know if whenever $$A$$ is an artinian ring, any $$A^n$$ also is and if so, how to prove it, because I can't find a simple answer to this question, since there is no easier caracterization as for noetherian rings that I know of. If we do have this, then $$M$$ is easily shown to be artinian by the existence of an epimorphism $$\pi : A^n \to M$$.

Can anyone give me a hint or something? Any help is appreciated :).

We know that a finite direct sum of artinian $$R$$-modules is artinian. Now each factor in $$R=A^n$$ is an artinian $$A$$-module by hypothesis, and a fortiori an artinian $$R$$-module.