Prove that a finite ring with which has no zero divisors is a field.
[We will assume Wedderburn’s Theorem.]
Let be a finite ring (not necessarily commutative) with having no zero divisors. Let be nonzero, and define by . Note that if , we have , and since , . Thus is injective. Since is finite, is surjective. Thus for some , we have . Similarly, with , there exists such that . By a previous theorem, is a unit in . Thus is a division ring.
By Wedderburn’s Theorem, is commutative; thus is a field.