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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 $$(1) \Longrightarrow (2)$$ ...
I am struggling with this proof so forgive me if my questions are possibly formulated badly ...Now in Bland's proof ... given that $$M$$ is finitely generated we have by Proposition 2.2.6 that ...
$$\exists \ f$$ such that $$f ( R^{ (n) } ) = M$$
for some homomorphism $$f$$ ... ... is that correct?... now ...
Bland argues that Corollary 4.2.6 shows that because $$f ( R^{ (n) } ) = M$$ then we have that $$M$$ is noetherian ... ...
... BUT ...
... how exactly do we use or employ Corollary 4.2.6 to show that $$f ( R^{ (n) } ) = M \Longrightarrow M$$ is noetherian ...What would $$M_1$$ and $$M_2$$ 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 $$(1) \Longrightarrow (2)$$ ...
I am struggling with this proof so forgive me if my questions are possibly formulated badly ...Now in Bland's proof ... given that $$M$$ is finitely generated we have by Proposition 2.2.6 that ...
$$\exists \ f$$ such that $$f ( R^{ (n) } ) = M$$
for some homomorphism $$f$$ ... ... is that correct?... now ...
Bland argues that Corollary 4.2.6 shows that because $$f ( R^{ (n) } ) = M$$ then we have that $$M$$ is noetherian ... ...
... BUT ...
... how exactly do we use or employ Corollary 4.2.6 to show that $$f ( R^{ (n) } ) = M \Longrightarrow M$$ is noetherian ...What would $$M_1$$ and $$M_2$$ 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
Last edited: