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How do I find the direct limit:

$lim \frac{1}{n}\mathbb{Z}$?

thanks.

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    OP, have a look at this question and see if that's what you want: http://math.stackexchange.com/questions/2040/colimit-of-frac1n-mathbbz2011-09-21

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Your question, as stated, doesn't make sense. What are the maps $n^{-1}\mathbb{Z}\to (n+1)^{-1}\mathbb{Z}$? We do get natural maps $n^{-1}\mathbb{Z}\to m^{-1}\mathbb{Z}$ if $n|m$, though (what are they?), and so we can take the colimit of this diagram. You should be able to check that it is isomorphic to $\mathbb{Q}$ because the colimit will identify different ways of writing the same fraction. That is, $\frac{a}{b}\in b^{-1}\mathbb{Z}$ and $\frac{ka}{kb}\in (kb)^{-1}\mathbb{Z}$ will get identiified because the map $b^{-1}\mathbb{Z}\to(kb)^{-1}\mathbb{Z}$ will take $\frac{a}{b}$ to $\frac{ka}{kb}$, meaning they will represent the same element in $\operatorname{colim} n^{-1}\mathbb{Z}$.

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    @paul: The notation $\lim$ is sometimes used for colimits as well, though usually in the form $\displaystyle \underset{\longrightarrow}{\lim}$.2011-09-22