Rings in Which Elements Are Uniquely the Sum of an Idempotent and a Unit

An associative ring with unity is called clean if every element is the sum of an idempotent and a unit; if this representation is unique for every element, we call the ring uniquely clean . These rings represent a natural generalization of the Boolean rings in that a ring is uniquely clean if and only if it is Boolean modulo the Jacobson radical and idempotents lift uniquely modulo the radical. We also show that every image of a uniquely clean ring is uniquely clean, and construct several noncommutative examples.