Is $\mathbb Q \times \{0\}$ an integral domain?
I understand that $\mathbb Q \times \{0\}$ is a commutative ring with unity. But there was no clear proof that it has no zero divisor.
How do I prove $\mathbb Q \times \{0\}$ is a zero divisor, if it's true that $\mathbb Q \times \{0\}$ is an integral domain?
If $(a,0)\ne (0,0)$ and $(b,0)\cdot (a,0)=(ba,0)=(0,0)$, then $ba=0$ and $a\ne 0$, and therefore $b=0$.