I'm having difficulty answering the following simple (?) question:
Let $X = \mathbb{Z}^n$, and let $M$ be a submodule of $X$ closed under division - by which I mean that if $x \in X$ has $n x \in M$ for some integer $n$, then $x \in M$.
I want to know:
1) Is $M$ complemented in $X$? (That is, is $M$ a direct summand as $\mathbb{Z}$-module?) (The usual counter example showing the difference between $\mathbb{Z}$-modules and vector spaces $2\mathbb{Z} \subset \mathbb{Z}$ does not pass my criterion.)
2) If so, how do I find the complement? (I.e. produce from a basis for $M$ a basis for the complement.)
I am asking for a condition under which one can extend a linearly independent set in a free $\mathbb{Z}$-module to a basis, and I am guessing that that condition is that the span is closed under division in the way described above.
Consider the short exact sequence $$0\to M\to X \to X/M \to 0.$$ Your divisibility condition essentially says that $X/M$ is torsion free. Thus, $X/M$ is free and the sequence splits, i.e., $X\cong M \oplus X/M$.