Characterizations of special clean elements and applications

Abstract

We prove that special clean decompositions of a given element of a ring are in one-to-one correspondence with the set of solutions of a simple equation in a corner ring. We then derive ``constructive'' proofs that in many rings, regular elements are special clean by solving this equation in specific cases. Other applications, such as uniqueness of decompositions, are given. Many examples of special clean decompositions of 2-2 matrices found by this methodology are also presented.