Is R a finite ring?

Because, if it is then we can prove it as follows:

If x is a zero-divisor the proof is complete.

If x is not a zero-divisor then consider the finite set,

{a_1x,a_2x,...,a_nx} where a_i are the non-zero elements of the ring (assuming it is non-trivial).

Then, we can show none are equal to each other.

(Excercise).

Then by Dirichlet's Pigeonhole Principle, is an enumeration of all non-zero elements in R. Hence we can find an a_i such that a_i x = 1. Thus, x is a unit.