My definition of an artianian module is : A module is artinian if every decending chain of submodules terminates.(adsbygoogle = window.adsbygoogle || []).push({});

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.

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

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