Finite Ring e1

Claim:

In a finite Ring R is each element $$a \in \mathbb{R} \setminus \{0\}$$ either a zero divisor or an unit (meaning it has an inverse).

Proof:

If a is a zero divisor, it is trivial.

Let a be not a zero divisor.

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

There are $$m,~n \in \mathbb{N}$$ with m < n und $$a^m = a^n$$.

That implies:

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

If $$a$$ is not a zero divisor, it holds $$a^m \neq 0$$

That implies $$1-a^{n-m} = 0$$.

That implies:

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

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

q.e.d.