Example of ring Let $R$ be a ring.
Let $A$ be the set of units of $R$. Let $B$ be the set of zero divisors of $R$ and let $C$ be the set of nonzero elements of $R$ which are not units or zero divisors.
Is there an example of a ring $R$ where $A$, $B$ and $C$ are non empty?
 A: It's easy to find all three in $\mathbb Z\times \mathbb Z$.
Also $\mathbb Z[x]/(x^2)$ provides all types.
Another one would be the ring of continuous functions from the reals into the reals.
A: Take $R=\mathbb{Z}/n[X]$ where $n$ is the product of two odd primes.
A: Let me suggest some guiding principle: If a ring contains a field then there will be lot of units. If it is not contained any field there will be zero divisors.
Take the set of all real valued continuous functions defined on whole of the real line: This contains constant functions, that form a subring which is in fact a field, providing units.
Now  take a real number $b$: Construct three functions $f_b, g_b, h_b$  as below:
$f_b(x)= 0$ for $x\le b$ and $f_b(x)= x-b$  for $x\ge b $
$g_b(x) = x-b$ for $x\le b$ and $g_b(x) = 0$ for $x\ge b$
$h_b(x) = x-b$ for all $x$
ALl of them are continuous functions throughout the real line; and at any $x$ one of $f_b$ or $g_b$ vanishes, so $f_b(x) g_b(x)$ is the constant function zero.
They provide zero-divisors in this ring (one for each real number).
The function $h_b$ is neither a unit nor a zero divisor.
This provides an example where all the sets are non-empty (in fact all are uncountable)
