Show that a finite ring R with unity 1 ≠ 0 and no divisors of 0 is a division ring. (It is actually a field, although commutativity is not easy to prove. See Theorem 24.10.) In your proof, to show that a ≠ 0 is a unit, you must show that a "left multiplicative inverse" of a ≠0 in R is also a "right multiplicative inverse."

Data from Theorem 24.10

Every finite division ring is a field. 

Proof See Artin, Nesbitt, and Thrall [24] for proof of Wedderburn's theorem.