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What does is mean to say that $\mathbb{Z}$ has no torsion?

This is an important fact for any course?

Thanks, I heard that in my field theory course, but I don't know what it is.

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    Seen [this](http://en.wikipedia.org/wiki/Torsion_%28algebra%29)?2012-08-28

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It means that $\forall\,n,z\in\Bbb Z\,\,,\,n\neq 0\,\,,\,nz=0\Longrightarrow z=0$ And yes, it is a rather important notion in group theory in general.

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    Anyone understanding the group operation defined on $\,\Bbb Z\,$ is expected to know that $nz:=\stackrel{\text{n times}}{\overbrace{z+z+...+z}}$2012-08-29
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A group $G$ has torsion if there are elements $g\neq0\in G$ such that $ng=0$ for some $n\neq 0$ (depending on $g$). In $\mathbb Z$, we have that $ng\neq 0$ for all $n,g\neq 0$ and so $\mathbb Z$ is torsion free.

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@d555, you might want to know that the notion of torison is extremely important, for example if $A$ is finitely generated and abelian group, then it can be written as the direct sum of its torsion subgroup $T(A)$ and a torsion-free subgroup (but this is not true for all infinitely generated abelian groups). $T(A)$ is uniquely determined.

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    Thanks, you're right. These fact is interesting2012-08-31
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In general, given an $R$-module $M$, an element $m \in M$ is called a torsion element if there exists some non zero $r \in R$ such that $rm=0$. Here I denote the set of torsion elements of $M$ is denoted $\text{Tor}(M)$, although I have also seen it denoted by $T(M)$. If $\text{Tor}(M)=0$, then $M$ is said to be torsion free.

For any ring $R$, one can think of $R$ as an $R$-module over itself, where scalar multiplication is simply the ring multiplication. In your case you are looking at the ring $\mathbb{Z}$, which is certainly a $\mathbb{Z}$-module. So the torsion elements of $\mathbb{Z}$ would be the set $\text{Tor}(\mathbb{Z})$. If $a \in \text{Tor}(\mathbb{Z})$, then there exists some $n\in \mathbb{Z}$ such that $na=0$. Since $\mathbb{Z}$ is an integral domain (the prototypical one in fact), the equation $na=0$ forces either $n=0$ or $a=0$. Thus $\text{Tor}(\mathbb{Z})=\{0\}$.