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

[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 don't understand that why is S_n artinian and how we have proved the theorem.

 

[morphism]

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.

 

[mathwonk]

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, don't just quote some theorem.

 

[peteryellow]

Thanks morphism and mathwonk.