In this paper we give an inductive new proof of the Jordan canonical form of a nilpotent element in an arbitrary ring. If
is a nilpotent element of index n with von Neumann regular
, we decompose
with
a Jordan block of size n over a corner S of R, and
nilpotent of index <n for an idempotent e of R commuting with a. This result makes it possible to characterize prime rings of bounded index n with a nilpotent element
of index n and von Neumann regular
as a matrix ring over a unital domain.
