I got one problem from the textbook:
Prove that a finite ring with more than one element and no zero divisors is a division ring.
It's obvious that the zero element must exist in this ring, so "with more than one element" means this ring is not trivial.
And once I prove that the identity element 1 exists in this ring, then the rest of work is easy.
So my question is reduced to the existence of the identity element.