- #1
peteryellow
- 47
- 0
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.
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.