Rings in which elements are uniquely the sum of an idempotent and a unit that commute. (Q958120)

Throughout \(R\) is an associative ring with identity. A ring is called (uniquely) clean if every element is (uniquely) the sum of an idempotent and a unit. Starting from the notion of strongly clean ring, defined as a ring where every element is the sum of an idempotent and a unit that commute with each other, the authors consider the notion of uniquely strongly clean ring, defined as before but asking for uniqueness of writing. These rings include uniquely clean rings and they arise as triangular matrix rings over commutative uniquely clean rings. The authors establish various basic properties of such rings and give several interesting and illuminating examples.