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I am focused on Section 4.2: Noetherian and Artinian Modules and need some further help to fully understand the proof of part of Proposition 4.2.10 ... ...

Proposition 4.2.10 reads as follows:View attachment 8215In the above proof by Bland we read the following:

" ... ... Continuing in this way we obtain an ascending chain \(\displaystyle Y \subseteq Y \oplus Y' \subseteq Y \oplus Y' \oplus Y'' \subseteq\) ... ... "Can someone please explain in some detail exactly how/why \(\displaystyle Y \subseteq Y \oplus Y'\) and \(\displaystyle Y \oplus Y' \subseteq Y \oplus Y' \oplus Y''\) etc ...

Help will be appreciated ...

Peter

=========================================================================Definition 4.2.9 is relevant to the above post so I am providing the text of Definition 4.2.9 ... as follows ... View attachment 8216=========================================================================