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∈R and consider the map on R given by x↦ax. If this map is injective then it has to be surjective, because R is finite. Hence, 1=ax for some x∈R and a is a unit. If the map is not injective then there are u,v∈R, with u≠v, such that au=av. But then a(u−v)=0 and u−v≠0 and so a is a zero divisor.
For the noncommutative case, see this answer.