Define the operations $\oplus$ and $\odot$ on $\Bbb Z$ by $a\oplus b = a+b-1$ and $a\odot b = ab+a+b$. Prove that $(\Bbb Z,\odot,\oplus)$ is an integral domain.
Surely they first want me to prove its a ring, so I'll just take that for granted here.
I just want to show that it is an integral domain, given that it is a ring here:
So we need to show that $a\odot b=0\implies a=0$ or $b=0$ I imagine. Okay, so:
$ab+a+b=0$, then if $ab=0$, since $\Bbb Z$ is an integral domain, it follows that either $a=0$ or $b=0$ and we are done. So assume not, let $ab\ne 0$, then $a\ne 0$ and $b\ne 0$ and we get $ab+a+b=0 \implies a=b=-2$ as the only valid solution when $ab\ne 0$.
I.e we have shown:
$a\odot b=0 \implies a=b=-2$ or $ab=0\implies a=0$ or $b=0$. Doesn't that mean in the case that $a=b=-2$ that this isn't an integral domain?
We have zero divisor $-2\odot -2=4-4=0$