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I am reading "Introduction to Ring Theory" by P. M. Cohn (Springer Undergraduate Mathematics Series)
In Chapter 2: Linear Algebras and Artinian Rings we find Theorem 2.2 on Noetherian modules. I need help with some aspects of the proof.
Theorem 2 reads as follows:View attachment 3158
View attachment 3157
In the proof of $$(c) \Longrightarrow (d)$$ we read:
If $$N$$ is a submodule of $$M$$, let $$\mathscr{C}$$ be the collection of all finitely generated submodules and choose a maximal term $$N'$$ in $$\mathscr{C}$$.
If $$N' \subset N$$, we can adjoin an element to $$N'$$ to obtain $$N''$$ in $$\mathscr{C}$$ and properly containing $$N'$$, but this contradicts the maximality of $$N'$$.
Hence, $$N' = N$$ and this shows $$N$$ to be finitely generated. … … "
My questions regarding this particular argument are as follows:
1. Why do we need the condition $$N' \subset N$$ in order to be justified in adjoining an element to $$N'$$ to obtain $$N''$$?
2. Why does/how does contradicting the maximality of N' imply that $$N' = N$$? What about the possibility that $$N \subset N'$$?Hoping someone can help.
Peter
In Chapter 2: Linear Algebras and Artinian Rings we find Theorem 2.2 on Noetherian modules. I need help with some aspects of the proof.
Theorem 2 reads as follows:View attachment 3158
View attachment 3157
In the proof of $$(c) \Longrightarrow (d)$$ we read:
If $$N$$ is a submodule of $$M$$, let $$\mathscr{C}$$ be the collection of all finitely generated submodules and choose a maximal term $$N'$$ in $$\mathscr{C}$$.
If $$N' \subset N$$, we can adjoin an element to $$N'$$ to obtain $$N''$$ in $$\mathscr{C}$$ and properly containing $$N'$$, but this contradicts the maximality of $$N'$$.
Hence, $$N' = N$$ and this shows $$N$$ to be finitely generated. … … "
My questions regarding this particular argument are as follows:
1. Why do we need the condition $$N' \subset N$$ in order to be justified in adjoining an element to $$N'$$ to obtain $$N''$$?
2. Why does/how does contradicting the maximality of N' imply that $$N' = N$$? What about the possibility that $$N \subset N'$$?Hoping someone can help.
Peter
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