# Proving that elements in a finite ring are zero divisors or units AND deducing that every finite integral is a field [duplicate]

Prove that if R is a finite ring with an identity, then every nonzero element R is either a zero divisor or a unit. Deduce that every finite integral is a field.

Hint: Let x be a nonzero element of R that is not a zero divisor. Show that $x^n$ for some $n \in N$, and deduce from this that x must be a unit.

There exist $n>m$ with $x^n=x^m$ this implies that $x^m(x^{n-m}-1)=0$, if $x^{m-1}(x^{n-m}-1)\neq 0$ then $x$ is a divisor of zero .
If $x^{m-1}(x^{n-m}-1)=0$ and $m=1$ we deduce that $x^{n-m}=1$ and $x$ is a unit, if $m>1$ we proceed recursively by repeating the previous step.