Characterizations of clean elements by means of outer inverses in rings and applications

Abstract

We characterize clean elements in unital and general rings by means of outer inverses. Some special cases, such as both clean and unit-regular elements, or strongly clean elements, are discussed. As an application, we also derive new characterizations of strongly regular elements.

Outer inverses in semigroups belonging to the prescribed Green’s equivalence classes

“…In addition, we discuss the existence criteria and characterization of inner inverses belonging to the prescribed principal right, left, and quasi-ideals. We recover that Drazin's (b, c)-inverse and Mary's inverse along an element are essentially the same (see also [47,Proposition 1.4, Corollary 1.1], [48], [75]). Both are outer inverses that belong to a prescribed Green's H -class, and the only difference is how that class is represented.…”

“…Mary [44] defined and studied inverses along an element using one of the most powerful tools of the semigroup theory -Green's relations. They have also been used in his other studies [45][46][47][48][49]. However, to solve equations uniquely, Mary mainly emphasized relation H .…”

Characterizations of special clean elements and applications

“…In [29] it is proved that unit-regularity can be weakened to group-regularity: a ∈ R is unit-regular iff it admits a Von Neumann inverse z ∈ R # . In [28] this result is sharpened as follows: a ∈ R is special clean iff it admits a reflexive inverse z ∈ R # . However, the proof is not direct and relies on the (b, c)-inverse of Drazin [11].…”

“…Actually, Camillo and Khuruna proved in [4] that elements of a unit-regular ring hold a stronger form of cleanness called special cleanness in [1]. Special cleanness is indeed both a refinement of cleanness but also of unit-regularity, as shown in [28] Theorem 4.1: a is special clean with decomposition a = ē + u, aR ∩ ēR = {0} iff it satisfies a = ē + u = au −1 a.…”