On r-Clean Algebras

A ring R is said to be r-clean if each element is the sum of an idempotent and a regular element of R. An r-clean ring generalizes the concept of a clean ring by replacing units with regular elements. If R is a commutative ring, an R-algebra A is considered clean if each element is the sum of an idempotent and a unit of A. In this work, we define an r-clean algebra as a generalization of a clean algebra. Additionally, we define an r-endoclean module as a module whose endomorphism ring is r-clean. Furthermore, we present the relationship between the r-clean properties of the R-algebra A, its base ring, and the algebra itself as a module over the ring