- #1

Math Amateur

Gold Member

MHB

- 3,998

- 48

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 \(\displaystyle M_1 \neq N_1\) then \(\displaystyle M_1 + N_1 = M\) since \(\displaystyle N_1\) is a maximal submodule of \(\displaystyle M\). ... ... "

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

Peter