Every \(a\in R\) for a finite ring is either a unit or a zero divisor.
Solution
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.