Question: > 
> Prove that the only Boolean ring that is an integral domain is $Z/2Z$.
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I know that the definition of a Boolean ring is $a^2=a$ and that an integral domain is $ab=0$ either $a=0$ or $b=0$. But yet i still can not solve the problem any idea please.

Answer: 
Let $R$ be a Boolean ring and an integral domain. For each $r \in R$, $r(r - 1) = r^2 - r = 0$. Hence, for each $r \in R$, $r = 0$ or $r - 1 = 0$, i.e., $r$ is either $0$ or $1$. So $R$ is the ring with two elements, i.e., $\Bbb Z/2\Bbb Z$.
Let $a \in R$. $a^2 = a$, implies that $a^2 - a =0$, $a(a-1)=0$. either $a = 0$ or $a-1=0$, thus $a=1$. Therefore $R$ is $Z/2Z$.