One-sided central Drazin inverses

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 $x\in R$ such that $xa\in C\left(R\right)$, $x{a}^{n+1}=a^n$ for some $n\in \mathbb{N}$. The right central Drazin invertible elements can be defined similarly. It is shown that an element $a\in R$ 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.

Keywords:

• l.c.-Drazin inverse
• r.c.-Drazin inverse
• central Drazin inverse
• Drazin inverse
• Dedekind-finite ring

2010 Mathematics Subject Classifications:

• 15A09
• 16E50

Acknowledgments

This work was supported by the Natural Science Foundation of Jiangsu Province of China (No. BK20181406).

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work was supported by the Natural Science Foundation of Jiangsu Province of China [grant number BK20181406].