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I am focused on Section 4.2: Noetherian and Artinian Modules and need some help to fully understand Example 4 on page 108 ... ...

Example 4 reads as follows:

View attachment 6114

In the above text, I do not understand the proof that \(\displaystyle R\) is not left Noetherian ... I hope someone can clearly explain the way the proof works ...

To try to clarify my lack of understanding of the proof ... Bland, in the above text writes ... ..." ... ... On the other hand, \(\displaystyle \mathbb{Q}\) is not Noetherian when viewed as a \(\displaystyle \mathbb{Z}\)-module.

Hence if \(\displaystyle S_1 \subseteq S_2 \subseteq S_3 \subseteq\) ... ... is a non-terminating chain of \(\displaystyle \mathbb{Z}\)-modules of \(\displaystyle \mathbb{Q}\), then \(\displaystyle \begin{pmatrix} 0 & S_1 \\ 0 & 0 \end{pmatrix} \subseteq \begin{pmatrix} 0 & S_2 \\ 0 & 0 \end{pmatrix} \subseteq \begin{pmatrix} 0 & S_3 \\ 0 & 0 \end{pmatrix} \subseteq\) ... ... is a non-terminating chain of left ideals of \(\displaystyle R\) ... ... "

*What is the justification for regarding \(\displaystyle \mathbb{Q}\) as a \(\displaystyle \mathbb{Z}\)-module ... indeed, couldn't we have done this while trying to prove/justify that \(\displaystyle R\) was right Noetherian and ended up with a non-terminating chain of right ideals of \(\displaystyle R\) ... ? Why is \(\displaystyle \mathbb{Q}\) not Noetherian when viewed as a \(\displaystyle \mathbb{Z}\)-module ... ... ?Can someone please simply and clearly explain the proof of \(\displaystyle R\) not being left Noetherian ... ... ?*

**My questions are as follows:**Hope someone can help ...

Peter