Are $x \cdot 0 = 0$, $x \cdot 1 = x$, and $-(-x) = x$ axioms? Context: Rings.


  Are $x \cdot 0 = 0$ and $x \cdot 1 = x$ and $-(-x) = x$ axioms?


Arguably three questions in one, but since they all are properties of the multiplication, I'll try my luck...
I will assume this is in the context of rings (e.g., real numbers, integers, etc).  In this case, the axiom defining $0$ is that $x + 0 = x$ for all $x$.  $x*0 = 0$ is a result of this since we have $x*0 = x*(0+0) = x*0 + x*0$ which implies $x*0 = 0$ (canceling one of the $x*0$'s).

I am guessing that for the second one you mean $x*1 = x$.  This is a definition (axiom).

The third one is a consequence of the definition of $-x$ being the element such that $x + (-x) = 0$.  For then we have $(-x) + x$ is also zero so that $x$ is the negative of $-x$.
Answering what you asked, the question is poorly formed because you didn't specify the theory you are talking about. To do that, well, you must define symbols, deduction rules... and axioms.
Answering what you probably meant to ask, no. Those are properties that $-$ and $\times$ have in the $\mathbb Z$ set, not axioms. They can be deduced from the operations themselves.