Special Zero Divisor in Ring Let $R$ be a ring.
A non-zero $a\in R$ is a called a zero divisor if there exits a non-zero $b\in R$ such that $ab=0$.
$a\in R$ is called an idempotent if $a^2 =a$.
$a\in R$ is called a nilpotent if there exists $n\in \mathbb{N}$ such that $a^n=0$.
It is clear and well-known that both idempotent and nilpotent are zero divisor.  
Is there any other special type of zero divisor in a ring?
 A: You are looking for examples of zero divisors that are neither idempotent nor nilpotent. Perhaps the simplest example of such an element occurs in the ring $\mathbb Z/6$, the ring of integers modulo $6$, where $2$ is a zero divisor ($2\cdot 3=0$) yet it is not nilpotent, since no power of $2$ is divisible by $6$.
An example of a different nature occurs in matrix rings, such as the ring of $n\times n$ matrices with real coefficients. Then, any matrix with determinant $0$ is a zero divisor, and the vast majority of such matrices is neither idempotent nor nilpotent. In other words, another class of zero divisors are the singular matrices in any matrix ring.
A: A slight generalization of the idempotent case:
Any element $a\ne 0$ that is not a root of unity, but fulfils $a^n=a$ for some $n>1$, is a zero divisor:
Proof: $0 = a^n-a = a(a^{n-1}-1)$. Since by assumption $a$ is not a roof of unity, $a^{n-1}-1\ne 0$.
Edit: Thinking further about it, one can generalize this even more:
Every non-zero non-unit element $a$ that fulfils $a^n=ua$ for some unit $u$ and $n>1$ is nilpotent.
Proof: $0 = a^n - ua  = (a^{n-1}-u)a$. Now if we had $a^{n-1}=u$, then $a^{n-1}u^{-1}=1$, therefore $a$ would be a unit with $a^{-1} = a^{n-2}u^{-1}$. But by assumption, $a$ is not an unit, therefore $a$ is nilpotent.
