prime and semiprime ring Can someone help me to solve these problems ?

1:Let $R$ be a prime ring. How can we prove that if $a \in R$ commutes with every element from a nonzero left ideal $L $ of $R$, then $a \in Z(R)$? 

Is this still true if $R$ is semiprime?
My other question is :

2:Is $0$ the only idempotent in a domain without unity?

Thanks for your attention.

$1-)$ If $x\in R$, $r\in L$ then $xr\in L$ hence $a(xr)=(xr)a$. But, $ar=ra$; we get that $(ax-xa)r=0$, which is to say, $(ax-xa)L=0$ for all $x\in R$. Since $R$ is prime and $L\neq 0$, we conclude that $ax=xa$ for all $x\in R$, hence $a\in Z(R)$.  I thınk It is is false for semiprime ring.  We can think $M_2(\mathbb{Z_2}).$( I will add more detail  in my free time )$2-)$ IT is true any idempotent $a\neq 1$ in  domain is zero divisior.   You can see more detail in Herstein 1976. 
