Kategorien

# In a finite ring each element is either zero divisor or unit

## 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.