Noetherian and Artinian modules are direct sum of finitely many indecomposible modules

This question was asked in my assignment on Algebraic Geometry and I need help in solving the problem. The question is related to Artinian and Noetherian Modules, I am not good at solving problem of this particular topic.

Question is : (a) Prove that every artinian module is a direct sum of finitely many indecomposible modules.

(b) Every Noetherian module is a direct sum of finitely many indecomposible modules.

Attempt: (a) and (b) appear to be similar. So, I am posting attempt of (a) only.Let V be an artinian module and it is not direct sum of finitely many indecomposible modules. WLOG, $V= V_1 \oplus V_2$ and $V_2$ is decomposible, which means that $V_2$ is non empty and $V_2= V_3\oplus V_4$. There are further cases now (1) $V_3$ and $V_4$ can be decomposible or both indecomposible ; So, I have a chain condition $V_3 \subseteq V_2 \subseteq V$ . Similar ascending chain can be built for (b).

But I don't know what my next step should be ?

Can you outline a proof?

Solution 1:

You may have seen the descending chain, but maybe you overlooked the obvious ascending chain you can get at the same time.

Let $V_0=V$ and $V'_0=\{0\}$. Suppose $V$ is not a finite sum of indecomposables. Then we can recursively define $V_n$ and $V'_n$ by looking at a proper decomposition $V_{n-1}=A\oplus B$ and setting $V_n$ to be one of $A$ or $B$ that is not a finite sum of indecomposables, and setting $V'_n$ to be the sum of $V'_{n-1}$ and the one of $A$ or $B$ that wasn't selected.

Now: look for yourself what the properties of $\{V_i\mid i\in\mathbb N\}$ and $\{V'_i\mid i\in\mathbb N\}$ are! The answer should be clear now.

The moral of the story is that infinite sequences of summands are going to produce both infinitely increasing and infinitely decreasing sequences of modules.