Show that $R$ is an integral domain iff for all $ x, y, z\in R, xy = xz $ implies $y = z$. 
Suppose $R$ is a commutative ring. Show that $R$ is an integral domain iff for all
      $ x, y, z\in R, xy = xz $ implies $y = z$.


Proof:
$\Rightarrow $Let $x,y,z\in R$ such that $x(y-z)=0$ where $R$ is a commutative ring. Because $R$ is an integral domain, so if $x\neq 0$, then $y-z=0$. Thus if $y-z=0$, then $xy=xz$, hence $y=z$.
$\Leftarrow$ Suppose $R$ is a commutative ring. Let $x,y,z\in R$ such that $xy=xy$ and $y=z$. Then $xy-xz=x(y-z)=0$.  So, it is either $x=0$ or $y-z=0$, thus $R$ is an integral domain. 

Is this right? If not, can anyone give me a hit to write better one? Thanks

In first $y-z=0 \implies y=z$.
In converse, Let $xy=0$, then $xy=0=x0$ and by hypothesis it implies that $y=0$ , so no proper zero divisors, and hence an integral domain.
