Clean Semiprime<i>f</i>-Rings with Bounded Inversion

Abstract

Abstract An element in a ring is called clean if it may be written as a sum of a unit and idempotent. The ring itself is called clean if every element is clean. Recently,Anderson and Camillo (Anderson,D. D.,Camillo,V. (2002). Commutative rings whose elements are a sum of a unit and an idempotent. Comm. Algebra 30(7):3327–3336) has shown that for commutative rings every von-Neumann regular ring as well as zero-dimensional rings are clean. Moreover,every clean ring is a pm-ring,that is every prime ideal is contained in a unique maximal ideal. In the same article,the authors give an example of a commutative ring which is a pm-ring yet not clean,e.g.,C(ℝ). It is this example which interests us. Our discussion shall take place in a more general setting. We assume that all rings are commutative with 1.