Question: Problem is "Let $R$ be a commutative ring with more than one element. prove that If for every nonzero element $a$ of $R$ we have $aR=R$, then $R$ is a field." 
If unity exists in $R$, it is trival. But i have no idea to prove it.

Answer: 
Since $R$ has more than one element, it has a nonzero element. Let $a\in R$, $a\neq 0$. Then $a\in aR$. Let $e\in R$ such that $ae=ea=a$.

Let $r\in R$. There is some $s\in R$ such that $as=r$. Now,
$$re=er=eas=as=r$$
so $e=1$ in $R$.
Take any element $a\in R$. Then $aR=R$ which means that $a\in aR$, so there exists some $e\in R$ such that $ae=ea=a$.

Now take any $b\in R$. Since $b\in aR$, there exists some $b'$ such that $b=ab'$ which means that $$b=ab'\\
b=(ea)b'\\
b=e(ab')\\
b=eb$$

so $b=eb=be$ for all $b\in R$, in other words, $e=1$.