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.
Take a set $Y$ of the required cardinality, $x$ an element not in $Y$, and $S$ the algebra of some field $K$ with generators $\{x\}\cup Y$ and relations $a^2=x$ and $ab=ba$ for all $a,b\in S$.
Seriously, if you don't want to specify what your question is about, you can only expect trivial answers like this.
Let $S$ be your set with the proposed structures, and $x$ is the distinguished element. I can only assume that you require $x \in S$, and that $S$ is closed under the usual operations of $+$ and $\times$, since otherwise, property (1) makes little sense.
Your property (2) implies that your set $S$ can only consist of $\{x\}$ (a singleton set). Indeed, if $a \neq x$, then $a\cdot x$ should not be a multiple of $x$, but certainly $ax$ is a multiple of $x$. Thus $a=x$.
Thus, only for cardinality = $1$ will $S$ exist.
