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. References: 16
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