Splitting of short exact sequence of finitely generated $F_p[[X]]$ modules Let A, B, C be finitely generated $F_p[[X]]$ modules where $F_P$ is a finite field such that  $ 0 \longrightarrow A \longrightarrow B \longrightarrow C \longrightarrow 0 $ holds. Then under what conditions $ B \cong A \oplus C $ ?
 A: Note that $\mathbb F_p[[X]]$ is a DVR (i.e. a PID with one primed ideal),
so the situation is similar to studying modules over other DVR's or PID's.
Probably the most familiar PID is $\mathbb Z$, and $\mathbb Z$-modules are just
abelian groups.  So you question is similar to asking for conditions on
when a s.e.s of f.g. ab. gps.
 $$0 \to A \to B \to C \to 0$$
splits. 
As in user115654's answer, the standard criterion is that $C$ be torsion free
(and hence free, and hence projective).
Another criterion is if $A$ and $C$ are finite, of coprime orders.  But note
that there is no analogue of this criterion for $\mathbb F_p[[X]]$, as the latter
ring has only one prime ideal, and so f.g. torsion modules (which is the analogue of
being a finite ab. gp.) can't have coprime annihilators; their annihilators
just have to be powers of the maximal ideal $X \mathbb F_p[[X]]$.
There are no other general criteria; after all, in the ab. gp. case, if $C$ is not free or 
if $A$ and $C$ are finite but don't have coprime torsion, then it is easy to write down non-split short exact sequences.  
A: I assume you're looking for criteria for the sequence to be split exact (this is different from asking for an abstract isomorphism $B \cong A \oplus C$). One sufficient condition for the sequence to split is that $C$ is torsionfree, since $F_p[[x]]$ is a PID and $C$ is assumed finitely generated. Note: the dual condition, that $A$ is injective, doesn't seem to apply here, because the injectives $\neq 0$ over $F_p[[x]]$ are not finitely generated.
