Weakly-Abel rings and weakly exchange rings

摘要

Let R be an associative ring with identity. An element x∈R is said to be weakly exchange if there exists an idempotent e∈R such that e∈xR and 1−e∈(1−x)R or 1−e∈(1+x)R. The ring R is said to be weakly exchange if all of its elements are weakly exchange. In this paper an element-wise characterization is given, and it is shown that weakly-Abel weakly exchange rings are weakly clean. Moreover, a relation between unit regular rings and weakly clean rings is also obtained.