Definition 10.84.1. Let $M$ be an $R$-module. A *direct sum dévissage* of $M$ is a family of submodules $(M_{\alpha })_{\alpha \in S}$, indexed by an ordinal $S$ and increasing (with respect to inclusion), such that:

$M_0 = 0$;

$M = \bigcup _{\alpha } M_{\alpha }$;

if $\alpha \in S$ is a limit ordinal, then $M_{\alpha } = \bigcup _{\beta < \alpha } M_{\beta }$;

if $\alpha + 1 \in S$, then $M_{\alpha }$ is a direct summand of $M_{\alpha + 1}$.

If moreover

$M_{\alpha + 1}/M_{\alpha }$ is countably generated for $\alpha + 1 \in S$,

then $(M_{\alpha })_{\alpha \in S}$ is called a *Kaplansky dévissage* of $M$.

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