Lemma 10.85.2. Let $M$ be a countably generated $R$-module with the following property: if $M = N \oplus N'$ with $N'$ a finite free $R$-module, then any element of $N$ is contained in a free direct summand of $N$. Then $M$ is free.

Proof. Let $x_1, x_2, \ldots$ be a countable set of generators for $M$. We inductively construct finite free direct summands $F_1, F_2, \ldots$ of $M$ such that for all $n$ we have that $F_1 \oplus \ldots \oplus F_ n$ is a direct summand of $M$ which contains $x_1, \ldots , x_ n$. Namely, given $F_1, \ldots , F_ n$ with the desired properties, write

$M = F_1 \oplus \ldots \oplus F_ n \oplus N$

and let $x \in N$ be the image of $x_{n + 1}$. Then we can find a free direct summand $F_{n + 1} \subset N$ containing $x$ by the assumption in the statement of the lemma. Of course we can replace $F_{n + 1}$ by a finite free direct summand of $F_{n + 1}$ and the induction step is complete. Then $M = \bigoplus _{i = 1}^{\infty } F_ i$ is free. $\square$

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