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 $$M / M_1$$ is a simple R-module, $$M / M_1$$ is artinian and noetherian ... ... Can someone please explain why $$M / M_1$$ being a simple R-module implies that $$M / M_1$$ 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 $$M / M_1$$ is a simple R-module, $$M / M_1$$ is artinian and noetherian ... ... Can someone please explain why $$M / M_1$$ being a simple R-module implies that $$M / M_1$$ is artinian and noetherian ... ... ?Peter

It now occurs to me that the answer to my question is quite straightforward ... indeed ...$$M / M_1$$ is simple $$\Longrightarrow$$ only submodules of $$M / M_1$$ are $$\{ 0 \}$$ and $$M / M_1$$$$\Longrightarrow$$ only descending and ascending chains of submodules are finite ... that is terminate in a finite number of elements$$\Longrightarrow$$ $$M / M_1$$ 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