I found this question when I tried to figure out what kind of subgroups of a free abelian group behave just as well as in the finitely generated case.

Let $M$ be a free abelian group and $N$ a subgroup of $M$. Suppose that $M/N$ is a direct sum of cyclic groups. Can we always find a basis $\{x_i\}_{i\in I}$ of $M$ such that $N=\bigoplus_{i\in I}(N\cap\mathbb Zx_i)$?

My attempt: Let $\varphi$ denote the canonical projection $M\to M/N$. Define a property $P$ on elements of $M$: for any $x\in M$, $P(x)$ is true iff for any $y\in M$ and $n\in\mathbb Z$, $ny=x$ implies $y\in\ker\varphi$. Using Zorn's Lemma, we can find a maximal submodule $L$ of $M$ consisting of elements satisfying $P$. I want to show $L$ is a direct summand of $M$, but I got stuck here.