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I am reading Paul E. Bland's book, "Rings and Their Modules".

I am focused on Section 4.2: Noetherian and Artinian Modules and need some help to fully understand the proof of part of Proposition 4.2.14 ... ...

Proposition 4.2.14 reads as follows:

https://www.physicsforums.com/attachments/8237

https://www.physicsforums.com/attachments/8235

In the above proof by Bland we read the following:

"... ... Since \(\displaystyle M / M_1\) is a simple R-module, \(\displaystyle M / M_1\) is artinian and noetherian ... ...

Can someone please explain why \(\displaystyle M / M_1\) being a simple R-module implies that \(\displaystyle M / M_1\) is artinian and noetherian ... ... ?

Peter

I am focused on Section 4.2: Noetherian and Artinian Modules and need some help to fully understand the proof of part of Proposition 4.2.14 ... ...

Proposition 4.2.14 reads as follows:

https://www.physicsforums.com/attachments/8237

https://www.physicsforums.com/attachments/8235

In the above proof by Bland we read the following:

"... ... Since \(\displaystyle M / M_1\) is a simple R-module, \(\displaystyle M / M_1\) is artinian and noetherian ... ...

Can someone please explain why \(\displaystyle M / M_1\) being a simple R-module implies that \(\displaystyle M / M_1\) is artinian and noetherian ... ... ?

Peter

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