Noetherian Rings and Modules: Theorem 2.2 - Cohn - Section 2.2 Chain Conditions

In summary, the conversation discusses Section 2.2 of P.M. Cohn's book, Introduction to Ring Theory, which covers Artinian and Noetherian rings and modules. The conversation also includes a discussion on Theorem 2.2 and its proof, which involves the concept of maximal members in a collection of submodules. The conversation ends with a request for confirmation on the analysis and a thank you note for the help provided in understanding ring and module theory.
  • #1
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I am reading P.M. Cohn's book: Introduction to Ring Theory (Springer Undergraduate Mathematics Series) ... ...

I am currently focused on Section 2.2: Chain Conditions ... which deals with Artinian and Noetherian rings and modules ... ...

I need help with understanding a feature of the Theorem and its proof ...

Theorem 2.2 and its proof read as follows:https://www.physicsforums.com/attachments/4900
View attachment 4901Part (c) of the above theorem effectively states that if Part (b) holds then any non-empty collection of submodules in \(\displaystyle M\) has a maximal member ... ...

Now my understanding of the proof of \(\displaystyle \text{ (b) } \Longrightarrow \text{ (c) }\) (which I wish someone to confirm) implies that any non-empty collection of submodules of \(\displaystyle M\) may actually have several or, indeed, many maximal members ... that is members that are maximal submodules of \(\displaystyle M\) ... since, following the proof of \(\displaystyle \text{ (b) } \Longrightarrow \text{ (c) }\), we may start with different members of the collection \(\displaystyle \mathscr{C}\) and build different strictly ascending chains which may end up having different maximal submodules ... ...

Is my analysis correct ... ... ?

I would appreciate it if someone would confirm my analysis is correct ... and/or ... point out any errors or shortcomings ...

Hope someone can help ... ...

Peter
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in order for MHB readers to appreciate the definitions and context to Theorem 2.2 in Cohn, I am providing Cohn's brief introduction to Section 2.2 Chain Conditions ... which reads as follows:View attachment 4902
 
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  • #2
I am sorry, Peter, but I cannot follow this reasoning. Maybe it is best to follow the proof in the text. Or you can try this: suppose you have a non-empty collection $\mathscr{C}$ of submodules of $M$ that does not have a maximal member, what happens then?
 
  • #3
steenis said:
I am sorry, Peter, but I cannot follow this reasoning. Maybe it is best to follow the proof in the text. Or you can try this: suppose you have a non-empty collection $\mathscr{C}$ of submodules of $M$ that does not have a maximal member, what happens then?
Thanks for reminding me of this issue, Steenis ...

I just revisited the Cohn text ... and have resolved the issue ...

By the way ... your help in the past has been critical and crucial to my understanding of ring and module theory ...

So thank you ...!

Peter
 
  • #4
You are welcome, Peter, it helps me too.
 

Related to Noetherian Rings and Modules: Theorem 2.2 - Cohn - Section 2.2 Chain Conditions

1. What is the significance of Noetherian rings and modules in mathematics?

Noetherian rings and modules are important structures in mathematics because they allow for a more systematic study of algebraic objects. They have properties that make them easier to work with and understand, and they have applications in various areas of mathematics, including algebraic geometry and representation theory.

2. What does Theorem 2.2 in Cohn's book on Noetherian rings and modules state?

Theorem 2.2 in Cohn's book states that if a ring R is Noetherian, then every submodule of a finitely generated module M over R is also finitely generated. In other words, every submodule of a finitely generated module over a Noetherian ring is also finitely generated.

3. How does Theorem 2.2 relate to chain conditions in Noetherian rings and modules?

Theorem 2.2 is closely related to the ascending chain condition (ACC) and the descending chain condition (DCC) in Noetherian rings and modules. These chain conditions state that every ascending or descending chain of submodules eventually stabilizes. Theorem 2.2 shows that if the ring is Noetherian, then every submodule of a finitely generated module satisfies these chain conditions.

4. Can you give an example of a Noetherian ring that is not a finitely generated module?

Yes, one example is the ring of polynomials in infinitely many variables over a field. This ring is Noetherian, but it is not a finitely generated module over itself. This is because there is no finite set of generators that can generate all polynomials in infinitely many variables.

5. How does Theorem 2.2 in Cohn's book contribute to the overall understanding of Noetherian rings and modules?

Theorem 2.2 is an important result that helps to characterize Noetherian rings and modules. It shows that the property of being Noetherian is preserved under submodules, which is a useful property for studying these structures. This theorem also allows for the simplification of proofs and the development of more general results in Noetherian ring theory.

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