Let’s say we have a module $U$ and we know its composition series is unique. I’m trying to follow a proof that shows that then $U$ cannot be decomposed as $U= V\oplus W $ where $V$ and $W$ are strict submodules.
The proof says that if this were the case then we would have at least two composition series, obtained by taking first a composition series for $V$ and then one for $W$ and vice versa.
I’m assuming that what they mean here is that we have a composition series for $V$ $$0=V_0 \subsetneq V_1 \subsetneq … \subsetneq V_n=V$$ and a composition series for $W$ $$ 0=W_0 \subsetneq … \subsetneq W_m=W$$ that we can join these two series in two ways as
$$0\oplus 0 \subsetneq V_1\oplus 0 \subsetneq… \subsetneq V \oplus 0 \subsetneq V\oplus W_1 \subsetneq…\subsetneq V \oplus W $$
And the other where we decompose $W$ first and then $V$.
Now the author just says that these last two are composition series. But why is this necessarily the case. How can you show that $$(V_i \oplus W)/(V_{i-1} \oplus W)$$ is simple and likewise $$(V\oplus W_i)/(V\oplus W_{i-1})? $$
EDIT
I’m guessing the two series have when joining the two composition series isn’t necessarily a composition series but it can be at least refined to one. Is this then the jist of the argument?
If $M$ and $N$ are modules and $K\subseteq M$ is a submodule, then $(M\oplus N)/(K\oplus N)\cong M/K$. Indeed, there is a surjective homomorphism $f:M\oplus N\to M/K$ defined by $f(m,n)=m+K$, and it is easy to see that the kernel of $f$ is $K\oplus N$.
So, in your situation, $(V_i\oplus W)/(V_{i-1}\oplus W)\cong V_i/V_{i-1}$ is simple and similarly for $(V\oplus W_i)/(V\oplus W_{i-1})$.