Unit uniquely clean rings

We define and explore the class of unit uniquely clean rings (or just UnitUC for short), that is a common generalization of the classes of uniquely clean rings and strongly nil-clean rings; in particular, Abelian UnitUC rings are uniquely clean and UnitUC rings with nil Jacobson radical are strongly nil-clean. These rings also generalize the so-called UUC and CUC rings as defined by Cӑalugӑareanu-Zhou in Mediterranean J. Math. (2023), which are rings whose clean elements are uniquely clean. These rings also represent a natural expansion of the classical Boolean rings in the sense that a ring is UnitUC if, and only if, it is exchange and Boolean modulo the Jacobson radical. The behavior of UnitUC rings under group ring and matrix ring extensions is investigated as well. In addition, several non-trivial examples are provided to explain and delimit the obtained results.