If for every nonzero element $a$ of $R$ we have $aR=R$..
Solution 1:
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 $R$.
Solution 2:
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, $e=1$.