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

.