There is an easy argument which shows that a finite integral domain (commutative unital ring with no zero divisors) is a field. Here I wonder whether this result still stands if the term "unital" is dropped.
In other words, can a finite commutative ring with no zero divisors always contain a multiplicative identity? More generally, if this is true, can we even generalize Wedderburn's little theorem: every finite ring with no zero divisors is a field?