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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:

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 ...?

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