# My definition of an artianian module is : A module is artinian if

1. Aug 15, 2008

### peteryellow

My definition of an artianian module is : A module is artinian if every decending chain of submodules terminates.

Let A be a semisimple ring and M an A-module.
If M is a finite direct sum of simple modules then M is artinian.

Proof: Suppose that M= S_1+...+S_n where + denotes direct sum. We prove this by induction on n. For n=1 we have that M is artinian. Assume the result for n-1. Then
$S_1+...+S_{n-1}$ and S_n are artinian modules and so is M.

Can somebody help me with this proof because I dont understand that why is S_n artinian and how we have proved the theorem.

2. Aug 15, 2008

### morphism

Re: modules

A simple module is trivially artinian.

And there is a theorem that states if a module M has an artinian submodule N such that the quotient M/N is artinian, then M is artinian too.

3. Aug 15, 2008

### mathwonk

Re: modules

try proving that if both V/W and W are finite dimensional vector spaces then so is V. its the same sort of thing. really prove it with your bare hands, dont just quote some theorem.

4. Aug 16, 2008

### peteryellow

Re: modules

Thanks morphism and mathwonk.