Let $\mathcal A$ be any Grothendieck abelian category and $0 \neq M \in \cal A$ an object. It is true that $M$ admits a simple subquotient?
It is certainly true for $\mathcal A=R-Mod$ since $M$ contains a cyclic module and any (left)ideal is contained in a maximal.
Motivation: In the category of modules over a ring $R$ the following are equivalent for an object $M$.
1) $M$ is a (maybe infinite) direct sum of simple modules.
2) Every short exact equence $$0 \rightarrow M'\rightarrow M\rightarrow M''\rightarrow 0$$ splits.
I want to generalize this statement to any abelian category and the above seems crucial for $2) \Rightarrow 1)$.
Edit: Hanno Becker informed me, that there are abelian categories without any irreducible objects, see Jeremy Rickards answer to this question. I changed my question accordingly.