Commutative Ring with Unity and Prime Characteristic Let $D$ be a integral domain with characteristic $p>0$. It is easy to prove that $p$ is prime. 
Now, if $R$ is a commutative ring with unity and characteristic $p$, does $p$ be a prime number implies $R$ be a integral domain, or there is an counterexample? 

No. Take $x=(0,1)$ and $y=(1,0)$ in $\mathbb{Z}_p\times \mathbb{Z}_p$. You get $xy=0$.
