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I am reading An Introduction to Rings and Modules With K-Theory in View by A.J. Berrick and M.E. Keating (B&K).
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 $$M$$ is finitely generated, there is a minimal subset $$\{ x_0, \ ... \ ... \ , x_s \}$$ of $$M$$ such that
$$x_0 R + \ ... \ ... \ , x_s R + L = M. \ ... \ ... \ ... $$" My problem is as follows:
I cannot see exactly why there exists a minimal subset $$\{ x_0, \ ... \ ... \ , x_s \}$$ of $$M$$ such that
$$x_0 R + \ ... \ ... \ , x_s R + L = M$$. ... ... ... Can someone please demonstrate, rigorously and formally, that there exists a minimal subset $$\{ x_0, \ ... \ ... \ , x_s \}$$ of $$M$$ such that
$$x_0 R + \ ... \ ... \ , x_s R + L = M$$?
Question 2In the above text by Berrick and Keating, we read the following:"... ... Let $$S$$ be the set of submodules $$X$$ of $$M$$ that contain $$x_1 R + \ ... \ ... \ , x_s R + L$$ but do not contain $$x_0$$. It is obvious that $$S$$ is inductive ... ..." Can someone please explain exactly why $$S$$ is inductive ... ... ?Hope someone can help ...
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
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 $$M$$ is finitely generated, there is a minimal subset $$\{ x_0, \ ... \ ... \ , x_s \}$$ of $$M$$ such that
$$x_0 R + \ ... \ ... \ , x_s R + L = M. \ ... \ ... \ ... $$" My problem is as follows:
I cannot see exactly why there exists a minimal subset $$\{ x_0, \ ... \ ... \ , x_s \}$$ of $$M$$ such that
$$x_0 R + \ ... \ ... \ , x_s R + L = M$$. ... ... ... Can someone please demonstrate, rigorously and formally, that there exists a minimal subset $$\{ x_0, \ ... \ ... \ , x_s \}$$ of $$M$$ such that
$$x_0 R + \ ... \ ... \ , x_s R + L = M$$?
Question 2In the above text by Berrick and Keating, we read the following:"... ... Let $$S$$ be the set of submodules $$X$$ of $$M$$ that contain $$x_1 R + \ ... \ ... \ , x_s R + L$$ but do not contain $$x_0$$. It is obvious that $$S$$ is inductive ... ..." Can someone please explain exactly why $$S$$ is inductive ... ... ?Hope someone can help ...
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