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I need help with the proof of Lemma 1.2.21 ...

Lemma 1.2.21 and its proof reads as follows:

View attachment 6037Question 1In the above text by Berrick and Keating, we read the following:"... ... Since \(\displaystyle M\) is finitely generated, there is a minimal subset \(\displaystyle \{ x_0, \ ... \ ... \ , x_s \}\) of \(\displaystyle M\) such that

\(\displaystyle x_0 R + \ ... \ ... \ , x_s R + L = M. \ ... \ ... \ ... \)" My problem is as follows:

I cannot see exactly why there exists a minimal subset \(\displaystyle \{ x_0, \ ... \ ... \ , x_s \}\) of \(\displaystyle M\) such that

\(\displaystyle x_0 R + \ ... \ ... \ , x_s R + L = M\). ... ... ... Can someone please demonstrate, rigorously and formally, that there exists a minimal subset \(\displaystyle \{ x_0, \ ... \ ... \ , x_s \}\) of \(\displaystyle M\) such that

\(\displaystyle x_0 R + \ ... \ ... \ , x_s R + L = M\)?

__In the above text by Berrick and Keating, we read the following:"... ... Let \(\displaystyle S\) be the set of submodules \(\displaystyle X\) of \(\displaystyle M\) that contain \(\displaystyle x_1 R + \ ... \ ... \ , x_s R + L\) but do not contain \(\displaystyle x_0\). It is obvious that \(\displaystyle S\) is inductive ... ..." Can someone please explain exactly why \(\displaystyle S\) is inductive ... ... ?Hope someone can help ...__

**Question 2**Peter========================================================================B&K's definition of "inductive" is contained in section 1.2.18 ... ... . which reads as follows:https://www.physicsforums.com/attachments/6038

View attachment 6039