REGULAR ELEMENTS WHICH IS A SUM OF AN IDEMPOTENT AND A LEFT CANCELLABLE ELEMENT

Let M be a right R-module, and let a ∈ EndRM be unit-regular. If EndR(Ima) is an exchange ring and EndR(Kera) has stable rank one, it is shown that there exist an idempotent e ∈ EndRM and a left cancellable u ∈ EndRM such that a = e + u and aM ∩ eM =0 .