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Main Question or Discussion Point
I am reading Paul E. Bland's book, "Rings and Their Modules".
I am focused on Section 4.2: Noetherian and Artinian Modules and need some help to fully understand the proof of part of Proposition 4.2.5 ... ...
Proposition 4.2.5 reads as follows:
My questions are as follows:
Question 1
In the above text from Bland we read the following:
" ... ... Conversely, suppose that ##N## and ##M/N## are noetherian. Let
##M_1 \subseteq M_2 \subseteq M_3 \subseteq \ ... \ ... ##
be an ascending chain of submodules of ##M##. Then
##M_1 \cap N \subseteq M_2 \cap N \subseteq M_3 \cap N \subseteq \ ... \ ... ## ... ... "
My question is ... what about the case where all the ##M_i## fail to intersect with N ... is this possible? ... if so how does the proof read then ...?
Question 2
In the above text from Bland we read the following:
" ... ... If ##i \ge n## and ##x \in M_i## then ##x + N \in (M_i + N)/N = (M_n + N)/N## ... ... "
My question is ... why does ##x \in M_i \Longrightarrow x + N \in (M_i + N)/N## ... ... is it because ...
##x \in M_i \Longrightarrow x + 0_N + N \in (M_i + N)/N## ...
... and ##x + 0_N + N = x + N## ... ... ?
Hope someone can help ...
Peter
I am focused on Section 4.2: Noetherian and Artinian Modules and need some help to fully understand the proof of part of Proposition 4.2.5 ... ...
Proposition 4.2.5 reads as follows:
My questions are as follows:
Question 1
In the above text from Bland we read the following:
" ... ... Conversely, suppose that ##N## and ##M/N## are noetherian. Let
##M_1 \subseteq M_2 \subseteq M_3 \subseteq \ ... \ ... ##
be an ascending chain of submodules of ##M##. Then
##M_1 \cap N \subseteq M_2 \cap N \subseteq M_3 \cap N \subseteq \ ... \ ... ## ... ... "
My question is ... what about the case where all the ##M_i## fail to intersect with N ... is this possible? ... if so how does the proof read then ...?
Question 2
In the above text from Bland we read the following:
" ... ... If ##i \ge n## and ##x \in M_i## then ##x + N \in (M_i + N)/N = (M_n + N)/N## ... ... "
My question is ... why does ##x \in M_i \Longrightarrow x + N \in (M_i + N)/N## ... ... is it because ...
##x \in M_i \Longrightarrow x + 0_N + N \in (M_i + N)/N## ...
... and ##x + 0_N + N = x + N## ... ... ?
Hope someone can help ...
Peter
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