= Ext separation of finite-length modules
{c}
Partition the isomorphism classes of <simple modules> into two sets $\mathcal C_1,\mathcal C_2$, with $\operatorname{Ext}^1(S,T)=\operatorname{Ext}^1(T,S)=0$ whenever $S,T$ lie in different sets. Every <module> $M$ of finite <composition length> then has a unique decomposition $M=U_1\oplus U_2$ into its largest <submodules> whose <Jordan–Hölder factors> belong to the respective sets.
For a simple $S\in\mathcal C_1$ and a finite-length $N$ with factors in $\mathcal C_2$, induction through the long exact sequence of the <Ext functor> gives $\operatorname{Ext}^1(S,N)=0$. Induct on the length of $M$: take $0\to N\to M\to S\to0$, split $N=N_1\oplus N_2$, and suppose $S\in\mathcal C_1$. The quotient extension $0\to N_2\to M/N_1\to S\to0$ splits. The inverse image of its $S$-summand complements $N_2$ in $M$ and has only factors in $\mathcal C_1$. Finally, Hom between modules with factors in different sets is zero, since any nonzero image would have a factor in both sets. Projecting any proposed <submodule> to the opposite summand therefore proves maximality and uniqueness.
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