Jordan-Holder Theorem for Modules .... .... Another Two Questions ....

[Math Amateur]

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 further help to fully understand the proof of part of Proposition 4.2.16 (Jordan-Holder) ... ...

Proposition 4.2.16 reads as follows:
View attachment 8243
https://www.physicsforums.com/attachments/8244
Question 1Near the middle of the above proof (top of page 116) we read the following:

"... ... so \(\displaystyle M_1 \cap N_1\) is a maximal submodule of \(\displaystyle M_1\) and \(\displaystyle N_1\), since \(\displaystyle M/M_1\) and \(\displaystyle M/N_1\) are simple modules. ... ... "Can someone please explain why \(\displaystyle M/M_1\) and \(\displaystyle M/N_1\) being simple modules implies that \(\displaystyle M_1 \cap N_1\) is a maximal submodule of \(\displaystyle M_1\) and \(\displaystyle N_1\) ... ... ?( ***NOTE*** : I can see that \(\displaystyle M/M_1\) and \(\displaystyle M/N_1\) being simple modules implies that \(\displaystyle M_1\) and \(\displaystyle N_1\) are maximal ... but how does that imply that \(\displaystyle M_1 \cap N_1\) is a maximal submodule of \(\displaystyle M_1\) and \(\displaystyle N_1\) ... ... ? ... ... )
Question 2Near the middle of the above proof (top of page 116) we read the following:

"... ... Using Proposition 4.2.14 we see that \(\displaystyle M \) is artinian and noetherian and Proposition 4.2.5 indicates that \(\displaystyle M_1 \cap N_1\) is artinian and noetherian ... ... "Can someone please explain how/why Proposition 4.2.5 indicates that \(\displaystyle M_1 \cap N_1\) is artinian and noetherian ... ...
Help will be appreciated ...

Peter
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The above post refers to Propositions 4.2.14 and 4.2.5 ... so I am providing text of the statements of the propositions as follows:View attachment 8245https://www.physicsforums.com/attachments/8246
Hope access to the above helps ...

Peter

 

[steenis]

(4.3) and (4.4) are compostion series of $M$, by definition $M_i$ is a maximal submodules of $M_{i-1}$, that is $ M_{i-1}/M_i$ is simple.
Therefore $M/M_1$ and $M/N_1$ are simple because $M = M_0 = N_0$.

In (4.7), since $M/M_1$ is simple, the isomorphic $N_1/(M_1 \cap N_1)$ must be simple, therefore $(M_1 \cap N_1)$ is maximal in $N_1$, idem $(M_1 \cap N_1)$ is maximal in $M_1$, using (4.8).

$(M_1 \cap N_1)$ is a submodule of $M$, and prop.4.2.5 says that submodules of noetherian modules are noetherian, idem arterian.