Semisimple quotient modules

Question

I am trying to show quotients of semisimplens modules need be semi simple but without use of correspondence. I am stuck on a minor (but major) detail:

(note, I showed submodules of semi simple need be semi simple)

Let $M$ be a semi simple module, let $N \subseteq M$ be a submodule.

Then we can write $M= N \oplus N'$ where $N'$ is some other submodule of $M$

Then we wish to show $M/N$ is semi simple, but this boils to showing

$(N \oplus N') / N$ is semisimple, if I can show this is isomorphic to $N'$,

Then Im done as $N'$ is a submodule and it is semi simple. How can I show

$(N \oplus N') / N \simeq N'$ ? I tried writing the short exact seq:

$\{0\} \rightarrow N \rightarrow N \oplus N' \rightarrow (N \oplus N') / N \rightarrow \{0\}$ But couldn't gather any info from here. help? hints?

Answer 1

Try naming a map $N \oplus N' \to N'$. What's the image? What's the kernel?