How is $\Bbb Z_0 = \{0, \pm m, \pm2m, \pm3m, \ldots\}$ denoted in set builder notation? $\Bbb Z_0$ has integers as its elements, and its elements are listed as follows:
$\Bbb Z_0 = \{0, \pm m, \pm2m, \pm3m, \ldots\}$
Then how is $\Bbb Z_0$ denoted in set builder notation?
Is there a correct representation for the set $\Bbb Z_0$ between the following two?
$\Bbb Z_0=$ {$x \in \Bbb Z\,|\, x=km \text{ for some } k \in \Bbb Z\ \text{for all integer m}$}  
OR 
$\Bbb Z_0=$ {$x \in \Bbb Z\,|\, x=km \text{ for some } k \in \Bbb Z\ \text{for some integer m}$}

A simple way would be to write it $m\Bbb Z=\{mn\in\Bbb Z:n\in\Bbb Z\}$.
