Document Type : Original Manuscript


Department of Mathematics, Damascus University, Damascus, Syria.


Abstract.An associative ring R with identity is called r¡clean ring if every
element of R is the sum of a regular and an idempotent element. In this paper,
we introduce the concept of r-clean rings relative to right ideal. We study
various properties of these rings. We give some relations between r-clean
rings and r-clean rings of 2 2 matrices over R relative to some right ideal
P. New characterization obtained include necessary and sufficient conditions
of a ring R to be r-clean in terms of P-regular, P-local and P-clean rings.
Also, We prove that every ring is r-clean relative to any maximal right ideal
of it.


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