On Commutative Clean Rings and pm Rings preprint version

Abstract. Rings are assumed to be commutative. Recent work gives some of the tools needed to characterize clean, almost clean, weakly clean and uniquely clean rings by describing their Pierce sheaves. The sheaf descriptions are used to show that weakly clean and almost clean rings which are pm rings are clean. A subclass of clean rings, here called J-clean rings, also known as F-semiperfect rings, is studied. It includes the uniquely clean rings. There is a mono-functor from commutative rings to J-clean rings which satisfies a universal property. Earlier non-functorial ways of embedding rings in J-clean rings can be derived from the functor. Applications to rings of continuous functions are found throughout. 1. Definitions and preliminaries. 1.1. Introduction. Throughout “ring ” will mean a commutative ring with 1 except in the first part of Section 4 where general unitary rings will make an appearance. Various authors have studied clean rings and related conditions. The following definition is a composite