The Stacks project

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:

1. $M_0 = 0$;

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

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

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

If moreover

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

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