Every element of a finite ring is a unit or a zero-divisor

Let \(a\in R\) and define \(\phi(x) = ax\).

If \(\phi\) is injective, then it is surjective, so \(1 = ax\) for some \(x \implies \inverseof{x} = a\).

Otherwise, \(ax_1 = ax_2\) with \(x_1 \neq x_2 \implies a(x_1 - x_2) = 0\) and \(x_1 - x_2 \neq 0\)

So \(a\) is a zero divisor.