Showing that a divisor of zero in a commutative ring with unity can have no multiplicative inverse" 
A divisor of zero in a commutative ring with unity can have no multiplicative inverse.

I don't understand why this statement is true.
So for $a,b$ in the ring, $ab=0$ by zero divisor.
How can we guarantee that there does not exist $c$ such that $ac=1$?

Hint Suppose $a$ has a multiplicative inverse, and multiply both sides of $ab = 0$ by $a^{-1}$.
