Lemma 10.84.3. Let $M$ be an $R$-module. Then $M$ is a direct sum of countably generated $R$-modules if and only if it admits a Kaplansky dévissage.

Proof. The lemma takes care of the “if” direction. Conversely, suppose $M = \bigoplus _{i \in I} N_ i$ where each $N_ i$ is a countably generated $R$-module. Well-order $I$ so that we can think of it as an ordinal. Then setting $M_ i = \bigoplus _{j < i} N_ j$ gives a Kaplansky dévissage $(M_ i)_{i \in I}$ of $M$. $\square$

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