On left r-clean bimodules

‎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.‎

Keywords:

left r-clean , left regular element , R-bimodule , left idempotent