@article { author = {Hakmi, H. and Alussein, B.}, title = {r-CLEAN RINGS RELATIVE TO RIGHT IDEALS}, journal = {Journal of Algebraic Systems}, volume = {10}, number = {2}, pages = {203-224}, year = {2023}, publisher = {Shahrood University of Technology}, issn = {2345-5128}, eissn = {2345-511X}, doi = {10.22044/jas.2021.10627.1525}, abstract = {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.}, keywords = {(P-)idempotents,Regular and P-regular rings,clean and r-clean rings,r-clean ring relative to right ideal}, url = {https://jas.shahroodut.ac.ir/article_2470.html}, eprint = {https://jas.shahroodut.ac.ir/article_2470_45b586cbdfdebd0f60d642784ff46ddd.pdf} }