I am trying to prove that the direct product $M = \mathbb{Z} \times \mathbb{Z}\times \cdots$ is not a projective $\mathbb Z$-module and I am stuck near the end of the proof because of the authors use of the word infinitely divisible.
I will list the sketch of the proof and try to explain where I am stuck.
We proceed by contradiction so suppose $M$ is contained in some free module $F$ with basis $B$. Set $N = \mathbb{Z} \oplus \mathbb{Z} \oplus\cdots$ and observe that $N$ is a submodule of $M$.
Since $N \subset F$ there exists $B' \subset B$ such that $B'$ is a basis for $N$ and consider the free module $F' \subset F$ determined by $B'$.
Notice $F'+M \subset F$ gives $M/(M \cap F') \cong (F'+M)/F' \subset F/F'$ so we have $M/(M \cap F') $
The next step in the proof requires to consider sequences of signs so let $s = (s_1, s_2, \ldots)$ be a sequence of plus and minus signs and consider an element $m_s := (s_1 , 2 s_2, \ldots , k! s_k, \ldots) \in M$
The next point in the proof is what I don't understand, the notes I am using say $m_s +(M \cap F')$ is infinitely divisible in $F/F'$ and use this to show $M$ cannot be contained in any free $Z$ module. My question is
How do we show $m_s +(M \cap F')$ is infinitely divisible in $F/F'$ and how do translate the word infinitely divisible into definitions from Hungerford or Dummite and Foote?