= Module of integer sequences
{title2=$\prod_{j\in\mathbb N_0}\mathbb Z$}
The module of integer sequences consists of all <functions> $\mathbb N_0\to\mathbb Z$, with pointwise addition and integer scalar multiplication. Its <submodule> of finitely supported sequences is the <direct sum of modules> $\bigoplus_{j\geq0}\mathbb Z$, a <free module> with coordinate <basis of a module> $(e_j)$. The full product is not a <free module>.
To prove this, suppose it had a <basis of a module> $E$. The <countable basis support lemma> places every finitely supported sequence in a <submodule> $C$ generated by a <countable> subset of $E$. The <basis partition gives a module direct sum> lemma supplies a projection $\pi$ onto its complementary <direct summand>, with <kernel> $C$. By <countability of countably generated abelian groups>, $C$ is <countable>. The <Cantor diagonal argument> makes the family of all sequences $w(j)=\pm2^j$ <uncountable>, so choose one with $\pi(w)\ne0$. Truncating $w$ before coordinate $i$ writes $w=f_i+2^ig_i$, where $f_i$ is finitely supported and $g_i$ is an integer sequence. Thus $\pi(w)=2^i\pi(g_i)$ for every $i$. The <vanishing of infinitely divisible integer sequences> forces $\pi(w)=0$, a contradiction.
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