## Assertion

In a finite ring R, each element \(a \in \mathbb{R} \setminus {0}\) is either a zero divisor or a unit (i.e. it has an inverse).

## Proof

If *a* is a zero divisor then we are done.

Let *a* be no zero divisor.

Since R is finite, the set $$\{a^n~|~n \in \mathbb{N}\}$$ is finite.

There is $$m,~n \in \mathbb{N}$$ with *m* < *n* and $$a^m = a^n$$

From this follows:

$$a^m = a^n\\ a^m – a^n = 0\\\ a^m\cdot (1-a^{n-m})=0$$

Since *a* is not a zero divisor, $$a^m \neq 0$$

Thus must be $$1-a^{n-m} = 0$$

From this follows:

$$1-a^{n-m} = 0\\\ 1 = a^{n-m}\\ 1 = a\cdot a^{n-m-1}$$

That is, $$a^{n-m-1}$$ is the inverse of *a* and *a* is a unit.

q.e.d.