University of Guilan Journal of Algebra and Related Topics 2345-3931 11 2 2023 12 01 On left r-clean bimodules 1 19 7404 10.22124/jart.2023.24516.1525 EN D. A. Yuwaningsih Department of Mathematics, Universitas Gadjah Mada , Yogyakarta, Indonesia Department of Mathematics Education, Universitas Ahmad Dahlan, Bantul, Indonesia I. E. Wijayanti Department Mathematics Universitas Gadjah Mada, Yogyakarta, Indonesia B. Surodjo Department of Mathematics, Universitas Gadjah Mada, Yogyakarta, Indonesia Journal Article 2023 05 13 ‎Let $R$ be an associative ring with identity and $M$ an $R$-bimodule‎. ‎We introduce the generalization of $r$-clean rings called left $r$-clean $R$-bimodules‎, ‎defined without their endomorphism rings‎. ‎An $R$-bimodule $M$ is said to be left $r$-clean if each element is the sum of a left idempotent and a left regular element of $M$‎. ‎We present some properties of the left $r$-clean $R$-bimodule‎. ‎At the end of this paper‎, ‎we give the sufficient and necessary condition for an $R$-bimodule to form a left $r$-clean $R$-bimodule.‎ https://jart.guilan.ac.ir/article_7404_6f51e4a4167b6e8b1f9ad7f7bf22ce37.pdf