We have of course:
$2\Bbb Z-1$, or $2\Bbb N+1$
These I think of as the integers reduced by the congruence $x\cong x+1$ where in one case $x$ is even and in the other $x$ is odd (it makes no difference really).
But I want to think specifically of reducing integers by the congruence $x\cong 2x$ perhaps more clearly the transitive closure of the equivalence relation $x\sim 2x$.
My random stab in the dark is to write it $\Bbb Z/\langle 2\rangle$
Expressed like this, I guess the set includes $0$.
Since this is a multiplicative group modulo a prime, perhaps it's better to exclude $0$ something like: $\Bbb Z^\times/\langle2\rangle$?
What's acceptable / usual here?