Ok. Let R be the ring. Since R has more than one element, take . Now since R is finite, so is . Since is finite and closed under multiplication, take some element and (This is because no zero divisors imply cancellation laws, so . so There exists a . Now for any element Since, no zero divisors imply cancellation laws are present, . Since say . So Multiply both right side by y, you get which means . Since y was any element from . the element is the identity.
As for Inverses, since for any there exists some such that . multiply left by , you get and so since , we get , so thus x has an inverse in .