Corollary 3.1.3 - Berrick and Keating - Noetherian Modules

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In summary, Corollary 3.1.3 states that for any direct sum $M \oplus N$ with generators $\{m_1, \dots, m_k\}$ for $M$ and $\{n_1, \dots, n_r\}$ for $N$, the set $\{(m_1,0), \dots, (m_k,0), (0,n_1), \dots, (0,n_r)\}$ generates $M \oplus N$. This is a result of Proposition 3.1.2, which states that for any module $M$ with generators $\{m_1, \dots, m_k\}$, the set $\{(m_1
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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 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
 
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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 $\Bbb Z \oplus \Bbb Z$.
 
  • #3
Deveno said:
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
 
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  • #4
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 $\beta(m,n) = n$.
 

FAQ: Corollary 3.1.3 - Berrick and Keating - Noetherian Modules

What is Corollary 3.1.3 in the context of Berrick and Keating's work?

Corollary 3.1.3 in Berrick and Keating's work states that every Noetherian module has a finite composition series, meaning it can be broken down into a finite sequence of submodules, each of which is a maximal submodule of the next. This is an important result in the study of Noetherian modules.

How does Corollary 3.1.3 relate to Noetherian modules?

Corollary 3.1.3 is a key theorem in the study of Noetherian modules, as it shows that every Noetherian module can be decomposed into a finite sequence of submodules. This allows for a better understanding and analysis of Noetherian modules.

Can you provide an example of Corollary 3.1.3 in action?

Yes, for example, consider a vector space over a field. This vector space can be seen as a Noetherian module, and Corollary 3.1.3 tells us that it can be decomposed into a finite sequence of subspaces, each of which is a maximal subspace of the next. This illustrates the usefulness of the corollary in breaking down complicated structures into simpler components.

Is Corollary 3.1.3 a widely accepted result in the field of Noetherian modules?

Yes, Corollary 3.1.3 is a well-established and widely accepted result in the field of Noetherian modules. It has been referenced and used in numerous other works and is considered an important theorem in the subject.

How does Corollary 3.1.3 impact other areas of mathematics?

Corollary 3.1.3 has implications beyond just the study of Noetherian modules. It has connections to other areas of mathematics, such as commutative algebra and algebraic geometry. In particular, it is useful in the study of algebraic varieties and their properties.

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