Composition Series and Noetherian and Artinian Modules ....

In summary, Peter is reading Paul E. Bland's book, "Rings and Their Modules" and is focused on Section 4.2: Noetherian and Artinian Modules. He needs help understanding the proof of Proposition 4.2.14, which states that M / M_1 being a simple R-module implies that it is both artinian and noetherian. Peter asks for an explanation, and it is clarified that this is because only finite ascending and descending chains of submodules are possible in a simple module. Peter thanks the responder and acknowledges that this is not true for rings.
  • #1
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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
 
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  • #2
Peter said:
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
 
  • #3
Yes, it is correct. It means that every simple module is fingen.

It is not true for rings, though.
 
  • #4
steenis said:
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
 

1. What is a composition series for a module?

A composition series for a module is a chain of submodules that cannot be further reduced or refined. This means that each submodule in the series is a direct summand of the next submodule, and the last submodule in the series is the module itself.

2. How is a composition series related to Noetherian and Artinian modules?

A module is Noetherian if it satisfies the ascending chain condition, which means that every increasing chain of submodules eventually stabilizes. Similarly, a module is Artinian if it satisfies the descending chain condition, which means that every decreasing chain of submodules eventually stabilizes. A composition series for a module is a special case of these conditions, where the chain of submodules is both increasing and decreasing.

3. Can a module have more than one composition series?

Yes, a module can have multiple composition series. This is because a module can have multiple direct summands, and each direct summand can be used as the first submodule in a composition series. However, all composition series for a given module will have the same length, as the length of a composition series is a property of the module itself.

4. How can we determine if a module is Noetherian or Artinian?

A module is Noetherian if and only if it has a finite composition series, and it is Artinian if and only if it has a finite descending chain of submodules. Therefore, to determine if a module is Noetherian or Artinian, we can check if it has a finite composition series or a finite descending chain of submodules, respectively.

5. Are Noetherian and Artinian modules important in mathematics?

Yes, Noetherian and Artinian modules are important concepts in mathematics, particularly in the study of abstract algebra and commutative algebra. They have applications in fields such as algebraic geometry, representation theory, and number theory. These concepts also have connections to other areas of mathematics, such as topology and homological algebra.

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