Composition Series and Noetherian and Artinian Modules ....

[Math Amateur]

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

 

[Math Amateur]

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:






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

It now occurs to me that the answer to my question is quite straightforward ... indeed ...


\(\displaystyle M / M_1\) is simple


\(\displaystyle \Longrightarrow\) only submodules of \(\displaystyle M / M_1\) are \(\displaystyle \{ 0 \}\) and \(\displaystyle M / M_1\)


\(\displaystyle \Longrightarrow\) only descending and ascending chains of submodules are finite ... that is terminate in a finite number of elements


\(\displaystyle \Longrightarrow\) \(\displaystyle M / M_1\) is artinian and noetherian ...



Is that correct ... ?

Peter

 

[steenis]

Yes, it is correct. It means that every simple module is fingen.

It is not true for rings, though.

 

[Math Amateur]

Yes, it is correct. It means that every simple module is fingen.

It is not true for rings, though.

Thanks steenis ...

Appreciate your help ...

Peter