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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 4.2.11 ... ...

Proposition 4.2.11 reads as follows:View attachment 8217I need help with the Proof of \(\displaystyle (1) \Longrightarrow (2)\) ...

I am struggling with this proof so forgive me if my questions are possibly formulated badly ...Now in Bland's proof ... given that \(\displaystyle M\) is finitely generated we have by Proposition 2.2.6 that ...

\(\displaystyle \exists \ f\) such that \(\displaystyle f ( R^{ (n) } ) = M\)

for some homomorphism \(\displaystyle f\) ... ...

Bland argues that Corollary 4.2.6 shows that because \(\displaystyle f ( R^{ (n) } ) = M\) then we have that \(\displaystyle M\) is noetherian ... ...

... BUT ...

... how exactly do we use or employ Corollary 4.2.6 to show that \(\displaystyle f ( R^{ (n) } ) = M \Longrightarrow M\) is noetherian ...What would \(\displaystyle M_1\) and \(\displaystyle M_2\) be in this case ... ?

Hope someone can help ...

Peter

=======================================================================***NOTE***

The above post refers to Proposition 2.2.6 and also to Corollary 4.2.6 ... so I am providing the text of each ... as follows:

View attachment 8218

View attachment 8219

View attachment 8220

Hope access to the above text helps ... ...

Peter

I am focused on Section 4.2: Noetherian and Artinian Modules and need some help to fully understand the proof of part of Proposition 4.2.11 ... ...

Proposition 4.2.11 reads as follows:View attachment 8217I need help with the Proof of \(\displaystyle (1) \Longrightarrow (2)\) ...

I am struggling with this proof so forgive me if my questions are possibly formulated badly ...Now in Bland's proof ... given that \(\displaystyle M\) is finitely generated we have by Proposition 2.2.6 that ...

\(\displaystyle \exists \ f\) such that \(\displaystyle f ( R^{ (n) } ) = M\)

for some homomorphism \(\displaystyle f\) ... ...

*... now ...***is that correct?**Bland argues that Corollary 4.2.6 shows that because \(\displaystyle f ( R^{ (n) } ) = M\) then we have that \(\displaystyle M\) is noetherian ... ...

... BUT ...

... how exactly do we use or employ Corollary 4.2.6 to show that \(\displaystyle f ( R^{ (n) } ) = M \Longrightarrow M\) is noetherian ...What would \(\displaystyle M_1\) and \(\displaystyle M_2\) be in this case ... ?

Hope someone can help ...

Peter

=======================================================================***NOTE***

The above post refers to Proposition 2.2.6 and also to Corollary 4.2.6 ... so I am providing the text of each ... as follows:

View attachment 8218

View attachment 8219

View attachment 8220

Hope access to the above text helps ... ...

Peter

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