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I am reading the book "An Introduction to Rings and Modules with K-theory in View" by A.J. Berrick and M.E. Keating ... ...
I am currently focused on Chapter 3; Noetherian Rings and Polynomial Rings.
I need help with yet another aspect of the proof of Proposition 3.1.2.
The statement and proof of Proposition 3.1.2 reads as follows (pages 109-110):https://www.physicsforums.com/attachments/4873
https://www.physicsforums.com/attachments/4874I need help with an aspect of the above proof ... indeed, I have two questions ...Question 1
In the above text from B&K write the following:
" ... ... Conversely, consider a submodule $$N$$ of $$M$$. Let $$N' = N \cap \alpha M'$$ and let $$N''$$ be the image of $$N$$ in $$M''$$, so that there is an exact sequence ... ... "
My question is as follows:
How do we know such an exact sequence exists ... that is, how do we demonstrate, formally and rigorously, that such a sequence exists ... ... ?Question 2
In the above text we read:
" ... ... Since both $$N'$$ and $$N''$$ are finitely generated, so also is $$N$$. ... ... "
How can we demonstrate, formally and rigorously that $$N'$$ and $$N''$$ being finitely generated, imply that $$N$$ is finitely generated ... ...Hope someone can help with these two questions ... ...
Peter
I am currently focused on Chapter 3; Noetherian Rings and Polynomial Rings.
I need help with yet another aspect of the proof of Proposition 3.1.2.
The statement and proof of Proposition 3.1.2 reads as follows (pages 109-110):https://www.physicsforums.com/attachments/4873
https://www.physicsforums.com/attachments/4874I need help with an aspect of the above proof ... indeed, I have two questions ...Question 1
In the above text from B&K write the following:
" ... ... Conversely, consider a submodule $$N$$ of $$M$$. Let $$N' = N \cap \alpha M'$$ and let $$N''$$ be the image of $$N$$ in $$M''$$, so that there is an exact sequence ... ... "
My question is as follows:
How do we know such an exact sequence exists ... that is, how do we demonstrate, formally and rigorously, that such a sequence exists ... ... ?Question 2
In the above text we read:
" ... ... Since both $$N'$$ and $$N''$$ are finitely generated, so also is $$N$$. ... ... "
How can we demonstrate, formally and rigorously that $$N'$$ and $$N''$$ being finitely generated, imply that $$N$$ is finitely generated ... ...Hope someone can help with these two questions ... ...
Peter