One-sided central Drazin inverses

Abstract

ABSTRACT We study the one-sided version of central Drazin inverses. An element a in a ring R is said to be left central Drazin invertible if there is such that , for some . The right central Drazin invertible elements can be defined similarly. It is shown that an element is central Drazin invertible if and only if a is both left and right central Drazin invertible. Some well-known results on Drazin inverses and left invertible elements including the famous Kaplansky theorem in [Jacobson N. Some remarks on one-sided inverses. Proc Amer Math Soc. 1950;1:352–355] are generalized. As applications, we give a new characterization of Dedekind-finite rings from the point of view of one-sided central Drazin invertible elements. The Cline's formula on Drazin invertible elements is also generalized.