I'm trying to understand the proof below for the fact a semisimple object in an abelian category is characterized by having every subobject complemented.
The argument seems elementary yet I'm still struggling. Note $P\leq M$ are semisimple.
- $N\oplus P\lneq M\implies M$ has a simple factor not present in $N\oplus P$. Why? Why can't it be that e.g $M,N\oplus P$ have the same simple factor with different multiplicities but $N\oplus P$ additionally has some indecomposable factor? I feel implication is missing a step.
- $S\cap(N+P)=0\implies N\cap(S+P)=0$. Why? I don't understand why this implication is true.
By assumption, $M$ is a sum of simple submodules, say $M = \sum_i S_i$. If every simple module satisfied $S_i \subseteq N + P$, then we would have $N + P = M$. Since $N + P \subsetneq M$, it must be the case that some simple $S$ is not contained in $N + P$.
$S \cap (N + P) = 0$ means in particular that $S \cap N = 0$ and $S \cap P = 0$. Since we already know $N \cap P = 0$, the following sum is direct: $S \oplus N \oplus P$. Does this make it clear?