Let $A$ be a finitely generated torsion $\mathbb{Z}\_p[[X]]$-module, $B$ = { $x \in A$ such that $px=0$ } and $C=A/B$ where $\mathbb{Z}\_p$ denotes the $p$-adic integers. Given $ 0 \rightarrow B/pB \rightarrow A/pA \rightarrow C/pC \rightarrow 0$ is a short exact sequence of abelian groups. Now $A/pA, B/pB, C/pC$ are finitely generated $\mathbb{F}\_p[[X]]$-modules where $\mathbb{F}\_p$ denotes the finite field. How to prove $A/pA \cong B/pB \oplus C/pC$ as $\mathbb{F}\_p[[X]]$-modules $?$



Splitting as $\mathbb{F}_p[[X]]$-modules
Better not to prove it, as it is wrong. Writing $M[p] = \lbrace m \in M: pm = 0 \rbrace$ for the exact $p$-torsion of a module $M$, I understand your question as follows: *If* the exact sequence
$$0 \rightarrow A[p] \rightarrow A \rightarrow A/A[p] \rightarrow 0$$
remains exact modulo $p$ (which is not always the case: it is equivalent to $A[p] \cap pA = 0$), then this exact reduced sequence
$$0 \rightarrow (A[p])/p \rightarrow A/p \rightarrow (A/A[p])/p \rightarrow 0$$ splits. A counterexample is $A = \mathbb{Z}\_p[\![X]\!] /(pX)$ where the reduced sequence
$$ 0 \rightarrow X\mathbb{F}\_p[\![X]\!] \rightarrow \mathbb{F}\_p[\![X]\!] \rightarrow \mathbb{F}\_p \rightarrow 0$$
is exact, but does not split, nor is there even any chance to write the middle term as direct sum of the outer ones (as $\mathbb{F}\_p[\![X]\!]$-module).


Maybe the illusion that the statement might be true comes from the fact that it is true for $p$-torsion modules (in particular, pseudonull modules) and the standard modules $\bigoplus\_i\mathbb{Z}\_p[\![X]\!]/(p^{\mu\_i})\oplus\bigoplus\_j\mathbb{Z}\_p[\![X]\!]/(f\_j^{m\_j})$, as a direct check shows.


Computing a pseudo-isomorphism from the above $A$ to the standard module $\mathbb{Z}\_p[\![X]\!]/(p)\oplus\mathbb{Z}\_p[\![X]\!]/(X)$ and seeing where things go wrong might be a good exercise in Iwasawa theory, compatible to questions like [Dual of a module](https://mathoverflow.net/questions/152322/) , [Iwasawa algebra](https://mathoverflow.net/questions/154031/) , [Rank of a $ \mathbb{Z}\_{p}[[T]] $ module](https://mathoverflow.net/questions/150442/) or [Iwasawa invariants](https://mathoverflow.net/questions/150912/).


