Noetherian Modules .... Cohn Theorem 2.2 .... ....

[Math Amateur]

I am reading P.M. Cohn's book: Introduction to Ring Theory (Springer Undergraduate Mathematics Series) ... ...

I am currently focused on Section 2.2: Chain Conditions ... which deals with Artinian and Noetherian rings and modules ... ...

I need help with understanding an aspect of the proof of Theorem 2.2 ... ...Theorem 2.2 and its proof (including some preliminary relevant definitions) read as follows:

At the end of the above proof by Cohn we read the following:

" ... ... If $a_j \in N_{i_j}$ and $k = \text{ max} \{ i_1, \ ... \ ... \ , i_r \}$, then equality holds in our chain from $N_k$ onwards. ... ... "
Can someone please explain how/why $a_j \in N_{i_j}$ and $k = \text{ max} \{ i_1, \ ... \ ... \ , i_r \}$ implies that equality holds in our chain from $N_k$ onwards. ... ... ?Help will be appreciated ...

Peter

 

[member 587159]

I am reading P.M. Cohn's book: Introduction to Ring Theory (Springer Undergraduate Mathematics Series) ... ...

I am currently focused on Section 2.2: Chain Conditions ... which deals with Artinian and Noetherian rings and modules ... ...

I need help with understanding an aspect of the proof of Theorem 2.2 ... ...Theorem 2.2 and its proof (including some preliminary relevant definitions) read as follows:View attachment 222878
View attachment 222879
At the end of the above proof by Cohn we read the following:

" ... ... If $a_j \in N_{i_j}$ and $k = \text{ max} \{ i_1, \ ... \ ... \ , i_r \}$, then equality holds in our chain from $N_k$ onwards. ... ... "
Can someone please explain how/why $a_j \in N_{i_j}$ and $k = \text{ max} \{ i_1, \ ... \ ... \ , i_r \}$ implies that equality holds in our chain from $N_k$ onwards. ... ... ?Help will be appreciated ...

Peter

It suffices to show that $N_{k+1} \subseteq N_k$.

Take $n \in N_{k+1} \subseteq N = (a_1, \dots, a_r)$. Then $n = \sum \lambda_i a_i$ for elements $\lambda_i$. Now, $a_1, \dots, a_r$ are contained in $N_k$, because we have an increasing chain, and hence $n\in N_k$ as well, since modules are closed under linear combinations.

 

[Math Amateur]

It suffices to show that $N_{k+1} \subseteq N_k$.

Take $n \in N_{k+1} \subseteq N = (a_1, \dots, a_r)$. Then $n = \sum \lambda_i a_i$ for elements $\lambda_i$. Now, $a_1, \dots, a_r$ are contained in $N_k$, because we have an increasing chain, and hence $n\in N_k$ as well, since modules are closed under linear combinations.

Thanks for the help Math_QED ...

BUT ... just some clarifications ...

You write:

" ... ... Take $n \in N_{k+1} \subseteq N = (a_1, \dots, a_r)$. ... ... "

This means $N_{k+1} \subseteq N$ and also $N = (a_1, \dots, a_r)$ ... is that correct ...

But then surely $n$ may equal $\sum_{ i = 1}^t \lambda_i a_i$ where $t \lt r$ ... and so $a_1, \dots, a_r$ may not all be contained in $N_{k + 1}$ let alone $N_k$ ...

I am really puzzled as to exactly why $(a_1, \dots, a_r)$ are contained in $N_k$ ... ... what is the exact argument?

Can you clarify?

Peter

 

[member 587159]

Thanks for the help Math_QED ...

BUT ... just some clarifications ...

You write:

" ... ... Take $n \in N_{k+1} \subseteq N = (a_1, \dots, a_r)$. ... ... "

This means $N_{k+1} \subseteq N$ and also $N = (a_1, \dots, a_r)$ ... is that correct ...

But then surely $n$ may equal $\sum_{ i = 1}^t \lambda_i a_i$ where $t \lt r$ ... and so $a_1, \dots, a_r$ may not all be contained in $N_{k + 1}$ let alone $N_k$ ...

I am really puzzled as to exactly why $(a_1, \dots, a_r)$ are contained in $N_k$ ... ... what is the exact argument?

Can you clarify?

Peter

For your first question, yes that's exactly what it means, for your second question:

We can find coefficients such that $n = \sum_{i=1}^r \lambda_i a_i$, because $n \in (a_1 \dots, a_r)$. But $a_1 \in N_{i_1}$

and $k = \max\{i_1, \dots, i_r\}$, so $k \geq i_1$. This implies that $N_{i_1} \subseteq N_k$, because the chain is increasing, so $a_1 \in N_k$ and you can do the same thing for the other indices, obtaining that $a_1, \dots, a_r \in N_k$

EDIT: Maybe you don't know what I mean with $(a_1, \dots, a_r)$. This is the module generated by the elements $a_1, \dots a_r$. I.e., the smallest submodule that contains these elements.