= Abelian groups as integer modules
{title2=$\mathbb Z\text{-modules}\;\longleftrightarrow\;\text{abelian groups}$}
Every <abelian group> has exactly one unital <module> structure over the <integers>. For $n>0$, define $nm$ by adding $m$ to itself $n$ times, put $0m=0$, and put $(-n)m=-(nm)$. The <module> axioms force this action, so no additional choice is involved. Moreover, every <Abelian group homomorphism> respects it and is a <module homomorphism>. See https://www.math.uwaterloo.ca/~dmckinno/pmath347/Modules.pdf[David McKinnon's Modules, the integer-scalar example].
For example, in the <free module> $\mathbb Z^2$,
$$
3(2,-1)+(-2)(1,4)=(4,-11).
$$
In the <cyclic group> $\mathbb Z/6\mathbb Z$, the same repeated-addition rule gives $4\overline2=\overline8=\overline2$ and $3\overline2=\overline0$. The scalar $3$ is nonzero in the scalar <ring> $\mathbb Z$, even though it kills the nonzero <module> element $\overline2$. Thus the scalar <ring> and the <module> must be kept distinct.
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