Examples of strongly clean rings

Abstract

A ring is called strongly clean if each of its elements can be written as the sum of an idempotent and a unit which commute. In this paper, we deal with two questions: first, is the center of a strongly clean ring R strongly clean, and second, is the power series ring over a strongly clean ring R strongly clean? Both questions have a positive answer when R is optimally clean, which is slightly more than strongly clean. We show that in general the questions have a negative answer, by giving, for any integral domain k, a strongly clean ring R such that and is not strongly clean. Moreover, we show that the optimally clean property is not a necessary condition for the answer to be positive, by giving, for any integral domain k, a strongly clean ring R such that and is strongly clean, but R is not optimally clean.