Is there any set of mathematical objects that satisfies all of the following:
1) For each object $a$ in the set, $a^2$ is some multiples of $x$.
2) $ab$ is never multiples of $x$ where $a$ and $b$ are any different two objects in the set.
3) all objects commute - $ab = ba$ and $a+b = b+a$.
Does such set exists for all cardinality?
Edit: to specify, let objects be matrices.