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This question already has an answer here:

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.

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marked as duplicate by MooS, rschwieb abstract-algebra May 5 '17 at 9:36

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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.

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