I'm looking for examples (if there are such) of non-commutative rings without multiplicative identity which have the following properties:
1) finite with zero divisors
2) infinite with zero divisors
3) finite without zero divisors
4) infinite without zero divisors
I'll be grateful for any examples and hints. Thanks in advance.
As indicated in Jared's comment, one may find easy examples of 1), 2), and 4) by taking non-unital subrings (such as left ideals or two-sided ideals) of rings with identity.
The most interesting request here is 3). In fact, any finite nonzero associative ring $R$ (possibly without identity) without zero divisors is a field.
First, let's prove that $R$ in fact has an identity. Let $a \in R$ be a nonzero element. The function $f \colon R \to R$ defined by $\phi(x) = ax$ is injective, and since $R$ is finite, it's a bijection. Again, because $R$ is finite, this bijection must have finite order. Thus for some $d$, the function $\phi^d(x) = a^d x$ is the identity. It follows that $a^d \in R$ is an identity element.
At this point, it's a standard exercise to show that a finite ring with identity and no zero divisors is a division ring. (Hint: think about the function above for any $a \in R$.) And Wedderburn's little theorem states that any finite division ring is a field.