Math Amateur
Gold Member
MHB
- 3,920
- 48
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 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/4839
https://www.physicsforums.com/attachments/4840In the above text (at the start of the proof), Berrick and Keating write:" ... ... Suppose that $$M$$ is Noetherian. A submodule of $$M'$$ is isomorphic to a submodule of $$M$$, and so is finitely generated. ... ... "I have two questions ... ...
Question 1
How do we demonstrate, formally and rigorously, that there exists a submodule of $$M'$$ that is isomorphic to a submodule of $$M$$ ... ... ?Question 2
How, exactly (that is, formally and rigorously), do we know that the submodule of $$M'$$ (which is isomorphic to a submodule of $$M$$) is finitely generated ... (I know it sounds plausible ... but ... what is the formal demonstration of this fact) ... ..Hope someone can help ...
Peter
I am currently focused on Chapter 3; Noetherian Rings and Polynomial Rings.
I need help with 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/4839
https://www.physicsforums.com/attachments/4840In the above text (at the start of the proof), Berrick and Keating write:" ... ... Suppose that $$M$$ is Noetherian. A submodule of $$M'$$ is isomorphic to a submodule of $$M$$, and so is finitely generated. ... ... "I have two questions ... ...
Question 1
How do we demonstrate, formally and rigorously, that there exists a submodule of $$M'$$ that is isomorphic to a submodule of $$M$$ ... ... ?Question 2
How, exactly (that is, formally and rigorously), do we know that the submodule of $$M'$$ (which is isomorphic to a submodule of $$M$$) is finitely generated ... (I know it sounds plausible ... but ... what is the formal demonstration of this fact) ... ..Hope someone can help ...
Peter