Weakly clean rings

We study the class of weakly clean rings which were introduced in [15]. It is known that weakly clean rings are a subclass of exchange rings and that they contain clean rings as a proper subclass. In this paper we prove that weakly clean rings also contain some other important examples of exchange rings, such as π  -regular rings and C⁎C⁎-algebras of real rank zero. Further, we prove that many classes of weakly clean rings can be viewed as corners of clean rings. This, for example, implies that every π  -regular ring and every C⁎C⁎-algebra of real rank zero is a corner of a clean ring. Lastly, we study the question when the ideal extension of weakly clean rings is weakly clean, and we give an example of a non-weakly clean exchange ring, answering the question in [15].