Proof from Axioms for a Ring

Question

I'm trying to prove this following theorem:

If $x, y \in \mathbb{Z}$, use the cancellation law for $\mathbb{Z}$ to demonstrate that $xy = 0 \implies$ $x = 0$ or $y = 0$

The proof I came up with doesn't quite seem definitive enough. I know how to prove this without using the cancellation law, but this requirement seems to make things much more difficult.

So, clearly $0 = 0 \cdot x = 0 \cdot y$ $\forall x, y \in \mathbb{Z}$. So, we can clearly say that this holds for $x \neq 0$ and $y \neq 0$.

So, first we can write that $xy = 0 \cdot x$ for $x \neq 0$. So, by the cancellation law, $y = 0$. Similarily, we can write that $xy =0 \cdot y$ for $y \neq 0$, so $x = 0$ by the cancellation law.

It seems to me that we can write $xy = 0 \cdot a$ for any integer, so this doesn't quite "prove" anything, though in such a case we wouldn't be able to use the cancellation law, so that wouldn't at all be a pertinent fact.

How does this sound?

Answer 1

It seems to me that we can write $xy=0⋅a$ for any integer, so this doesn't quite "prove" anything,

And if $y\ne 0$ you can write $x = x*y*y^{-1} = 0*a*y^{-1} = 0*(a*y^{-1}) = 0$ for any integer. So that proves everything.

Answer 2

The cancellation law says that if $c\ne 0$ and $ac = bc$ then $a=b$.

If $xy=0$ and $x\ne 0$, then since $xy = x0$, $y=0$.