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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 $$M$$ has a maximal member ... ...
Now my understanding of the proof of $$\text{ (b) } \Longrightarrow \text{ (c) }$$ (which I wish someone to confirm) implies that any non-empty collection of submodules of $$M$$ may actually have several or, indeed, many maximal members ... that is members that are maximal submodules of $$M$$ ... since, following the proof of $$\text{ (b) } \Longrightarrow \text{ (c) }$$, we may start with different members of the collection $$\mathscr{C}$$ 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
=================================================
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
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 $$M$$ has a maximal member ... ...
Now my understanding of the proof of $$\text{ (b) } \Longrightarrow \text{ (c) }$$ (which I wish someone to confirm) implies that any non-empty collection of submodules of $$M$$ may actually have several or, indeed, many maximal members ... that is members that are maximal submodules of $$M$$ ... since, following the proof of $$\text{ (b) } \Longrightarrow \text{ (c) }$$, we may start with different members of the collection $$\mathscr{C}$$ 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
=================================================
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