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I am reading Paul E. Bland's book, "Rings and Their Modules".

I am focused on Section 4.3: Modules Over Principal Ideal Domains ... and I need some help to fully understand the proof of Lemma 4.3.10 ... ...

Lemma 4.3.10 and its proof read as follows:https://www.physicsforums.com/attachments/8283

View attachment 8284I do not follow the strategy of the proof ... for example Bland writes:

" ... ... Thus, \(\displaystyle xR \subseteq x_1 R\) and we claim that this containment is proper ... ... "But ... why is Bland proving that this containment is proper ... what is the point ... how is this furthering the proof ...?Peter

I am focused on Section 4.3: Modules Over Principal Ideal Domains ... and I need some help to fully understand the proof of Lemma 4.3.10 ... ...

Lemma 4.3.10 and its proof read as follows:https://www.physicsforums.com/attachments/8283

View attachment 8284I do not follow the strategy of the proof ... for example Bland writes:

" ... ... Thus, \(\displaystyle xR \subseteq x_1 R\) and we claim that this containment is proper ... ... "But ... why is Bland proving that this containment is proper ... what is the point ... how is this furthering the proof ...?Peter

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