r-CLEAN RINGS RELATIVE TO RIGHT IDEALS
An associative ring R with identity is called r¡clean ring if everyelement 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 studyvarious properties of these rings. We give some relations between r-cleanrings and r-clean rings of 2 2 matrices over R relative to some right idealP. New characterization obtained include necessary and sufficient conditionsof 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 idealof it.
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