# Composition Series and Noetherian and Artinian Modules ....

• MHB
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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 ... ...

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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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 ... ...

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

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

It is not true for rings, though.

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

It is not true for rings, though.

Thanks steenis ...