# Every nonzero element in a finite ring is either a unit or a zero divisor

Let $R$ be a finite ring with unity. Prove that every nonzero element of $R$ is either a unit or a zero-divisor.

In a finite commutative ring with unity, every element is either a unit or a zero-divisor. Indeed, let $$a∈Ra\in R$$ and consider the map on $$RR$$ given by $$x↦axx \mapsto ax$$. If this map is injective then it has to be surjective, because $$RR$$ is finite. Hence, $$1=ax1=ax$$ for some $$x∈Rx\in R$$ and $$aa$$ is a unit. If the map is not injective then there are $$u,v∈Ru,v\in R$$, with $$u≠vu\ne v$$, such that $$au=avau=av$$. But then $$a(u−v)=0a(u-v)=0$$ and $$u−v≠0u-v\ne0$$ and so $$aa$$ is a zero divisor.