ON r-CLEAN IDEALS

Let R be an associative ring with identity. Ring R is considered clean if each element is the sum of an idempotent element and a unit element. A clean ring is generalized to an r-clean ring by generalizing the unit elements to regular. A ring R is said to be r-clean if each element is the sum of an idempotent element and a regular element. We introduce the notion of the r-clean ideal, a natural generalization of the clean ideal. Furthermore, we present some properties of the r-clean ideal and offer several sufficient and necessary conditions for a clean ideal to be r-clean.