%0 Journal Article %T r-CLEAN RINGS RELATIVE TO RIGHT IDEALS %J Journal of Algebraic Systems %I Shahrood University of Technology %Z 2345-5128 %A Hakmi, H. Ibrahim %A Alussein, B. Ali %D 2023 %\ 01/01/2023 %V 10 %N 2 %P 203-224 %! r-CLEAN RINGS RELATIVE TO RIGHT IDEALS %K (P-)idempotents %K Regular and P-regular rings %K clean and r-clean rings %K r-clean ring relative to right ideal %R 10.22044/jas.2021.10627.1525 %X Abstract.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. %U https://jas.shahroodut.ac.ir/article_2470_45b586cbdfdebd0f60d642784ff46ddd.pdf