TITLE: Finite rings without zero divisors are division rings.
QUESTION [12 upvotes]: How can I prove this:  

Finite rings without zero divisors are division rings.

I know how to prove it when the ring has $1$, but I have no idea if my ring needs to have an unity.

REPLY [7 votes]: Consider, for $a\in R$, the map $r_a\colon R\to R$ defined by $r_a(x)=xa$; this map is injective if $a\ne0$, hence surjective. So there's an element $b$ such that $ba=a$. Of course $b\ne0$; therefore there exists $c\in R$ such that $bc=b$. Then, for each $x\in R$,
$$
bx=bcx
$$
and so $x=cx$. Therefore there is a left identity. Similarly there's a right identity.