MHB Corollary 3.1.3 - Berrick and Keating - Noetherian Modules

[Math Amateur]

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 Corollary 3.1.3.

The statement of Corollary 3.1.3 reads as follows (page 110):https://www.physicsforums.com/attachments/4875Now, Berrick and Keating give no proof of this Corollary: presumably they think the proof is simple and obvious. Maybe it is ... but I need help in formulating a formal and rigorous proof ... can someone please help ...Because we are dealing with a Corollary to Proposition 3.1.2 readers of this post need the text of that Proposition. Proposition 3.1.2 and its proof are as follows:https://www.physicsforums.com/attachments/4876
https://www.physicsforums.com/attachments/4877Hope someone can help with the proof of Corollary 3.1.3 ... ...

Peter

 

[Deveno]

Suppose $\{m_1,\dots,m_k\}$ generate $M$, and $\{n_1,\dots,n_r\}$ generate $N$.

Then $\{(m_1,0),\dots,(m_k,0),(0,n_1),\dots,(0,n_r)\}$ generate $M \oplus N$.

For example, $\{(1,0),(0,1)\}$ generates $\Bbb Z \oplus \Bbb Z$.

 

[Math Amateur]

Suppose $\{m_1,\dots,m_k\}$ generate $M$, and $\{n_1,\dots,n_r\}$ generate $N$.

Then $\{(m_1,0),\dots,(m_k,0),(0,n_1),\dots,(0,n_r)\}$ generate $M \oplus N$.

For example, $\{(1,0),(0,1)\}$ generates $\Bbb Z \oplus \Bbb Z$.

Oh OK ... thanks Deveno ... clear now ... appreciate your help ...

BUT ... clear as it is ... it does not seem to be a Corollary of Proposition 3.1.2 ... how does it depend on Proposition 3.1.2

Can you help with the second statement of the Corollary?

Peter

 

[Deveno]

For any direct sum $M \oplus N$ we clearly have the short exact sequence:

$0 \to M \to M \oplus N \to N \to 0$,

where $\alpha: M \to M \oplus N$ is given by $\alpha(m) = (m,0_N)$

and $\beta: M\oplus N \to N$ is given by $\beta(m,n) = n$.