Rings in Which Every Element is the Sum of a Unit and a Full Idempotent

We introduce the class of full-idempotent clean rings. A ring R is said to be full-idempotent clean if every element a ∈ R can be expressed as a = f + u, where u is a unit and f is either 0 or a full idempotent, that is, RfR = R. This notion naturally interpolates between local rings and clean rings, imposing stronger structural constraints than cleanness alone. In this work, we investigate the fundamental properties of fullidempotent clean rings, examining their behavior under ring homomorphisms, factor rings, matrix constructions, and other classical ring-theoretic operations. Our results provide a unified framework for understanding the interplay between cleanness and fullness in both commutative and noncommutative settings.