Does a finite commutative ring necessarily have a unity?
I ask because of the following theorem given in my lecture notes:
Theorem. In a finite commutative ring every non-zero-divisor is a unit.
If it had said "finite commutative ring with unity..." there would be no question to ask, I understand that part. What I'm asking about is whether or not we can omit explicitly stating it because it follows from the finiteness of our commutative ring.
[Clarification] The way I'm learning ring theory now, a "ring" is defined as an additive Abelian group further equipped (I hope I'm using the right terminology) with an associative multiplication operation which distributes over addition. In this definition we do not require the existence of 1.
In other words, when I say "ring" I mean a rng.