The Nicholson-Varadarajan Theorem on Clean Linear Transformations

Extract

An element r in a ring R is clean if r is a sum of a unit and an idempotent. Camillo and Yu showed that unit regular rings are clean and in a very surprising development Nicholson and Varadarajan showed that linear transformations on countable dimension vector spaces over division rings are clean. These rings are very far from being unit regular.

Here we note that an idempotent is just a root of g(x)=x^{2}-x. For any g(x) we say R is g(x)-clean if every r in R is a sum of a root of g(x) and a unit. We show that if V is a countable dimensional vector space and over a division ring D and g(x) is any polynomial with coefficients in <formtex>K={\text Center}D and two distinct roots in K, then {\text End}V_D is g(x)-clean.