Example of a nilpotent division ring Actually, my headline pretty much sums up my question. I have done some exercises on nilpotent group and I'm thinking about an example of a division ring $(D,+,\cdot )$ where $(D,+)$ is an abelian group and $(D\setminus 0,\cdot)$ is a nilpotent group. 
 A: If $x$ is a nilpotent element that is also a unit, then it is $0$.
This means that no nilpotent ring can have any nonzero invertible elements. 
A division ring has at least its identity as a nonzero invertible element.
So no, there isn't any such ring.

Update
The user has since updated their question to indicate they are using a homemade definition of "nilpotent ring" that is not consistent with the standard notion I was using.  The following is added after that came to light.
After doing a little research I came across this interesting survey:

Hazrat, Roozbeh, M. Mahdavi-Hezavehi, and Mehran Motiee. "Multiplicative groups of division rings." Mathematical Proceedings of the Royal Irish Academy. Vol. 114. No. 1. Royal Irish Academy, 2014.

In it, they mention that in 1950, Hua proved that when the multiplicative group of a division ring is solvable, the division ring is actually a field. Since nilpotent groups are solvable, this would say that all examples are actually fields.
Since finite fields are obvious examples of "nilpotent fields" in the sense you gave, I guess you were mainly interested in noncommutative examples, but the above account says that this is not possible.
