Jordan-Holder Theorem for Modules .... ....

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

Proposition 4.2.16 reads as follows:
View attachment 8240
View attachment 8241

Near the middle of the above proof (top of page 116) we read the following:

"... ... If $$M_1 \neq N_1$$ then $$M_1 + N_1 = M$$ since $$N_1$$ is a maximal submodule of $$M$$. ... ... "

Can someone please explain exactly how $$N_1$$ being a maximal submodule of $$M$$ implies that $$M_1 + N_1 = M$$ ... ... ?

Peter

 

[GJA]

Hi Peter,

Near the middle of the above proof (top of page 116) we read the following:

"... ... If $$M_1 \neq N_1$$ then $$M_1 + N_1 = M$$ since $$N_1$$ is a maximal submodule of $$M$$. ... ... "

Since $M_{1}\neq N_{1}$, $M_{1}+N_{1}$ is a submodule of $M$ containing the maximal submodule $N_{1}$ and so must be $M$.

 

[Math Amateur]

Hi Peter,
Since $M_{1}\neq N_{1}$, $M_{1}+N_{1}$ is a submodule of $M$ containing the maximal submodule $N_{1}$ and so must be $M$.

Thanks for the help, GJA ...

Peter