A Jordan canonical form for nilpotent elements in an arbitrary ring

Abstract

In this paper we give an inductive new proof of the Jordan canonical form of a nilpotent element in an arbitrary ring. If a∈R is a nilpotent element of index n with von Neumann regular an−1, we decompose a=ea+(1−e)a with ea∈eRe≅Mn(S) a Jordan block of size n over a corner S of R, and (1−e)a 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 a∈R of index n and von Neumann regular an−1 as a matrix ring over a unital domain.