Clean general rings

An associative ring with unity is called clean (respectively uniquely clean) if every element is (uniquely) the sum of an idempotent and a unit. In this paper we define clean general rings (with or without a unity) and extend many of the basic results to the wider class. In particular, a clean general ring is an exchange ring in the sense of Ara. We then study the general analogue of the uniquely clean rings and their relationship to the boolean rings. Finally, we introduce semiboolean rings as a natural generalization of boolean rings.