Suppose a is invertible. Now let b be an element such that ab = 0. Multiply the latter equality on both sides by a^-1 to obtain a^-1 * (ab) = 0. Using the associativity of multiplication gives b = (a^-1 * a) * b = 0, whence b = 0. In particular, a is not a zero divisor.

This is a trivial restatement of the definition of domain; the proof is left to the reader.