Turkish Journal of Mathematics
Abstract
In this paper, we investigate the elements whose Moore-Penrose inverse is idempotent in a ${\ast}$-ring. Let $R$ be a ${\ast}$-ring and $a\in R^\dagger$. Firstly, we give a concise characterization for the idempotency of $a^\dagger$ as follows: $a\in R^\dagger$ and $a^\dagger$ is idempotent if and only if $a\in R^{\#}$ and $a^2=aa^*a$, which connects Moore-Penrose invertibility and group invertibility. Secondly, we generalize the results of Baksalary and Trenkler from complex matrices to ${\ast}$-rings. More equivalent conditions which ensure the idempotency of $a^\dagger$ are given. Particularly, we provide the characterizations for both $a$ and $a^\dagger$ being idempotent. Finally, the equivalent conditions under which $a$ is EP and $a^\dagger$ is idempotent are investigated.
DOI
10.3906/mat-2101-93
Keywords
Moore-Penrose inverse, group inverse, core inverse, idempotent, EP
First Page
878
Last Page
889
Recommended Citation
ZHU, H, CHEN, J, & ZHOU, Y (2021). On elements whose Moore-Penrose inverse is idempotent in a ${\ast}$-ring. Turkish Journal of Mathematics 45 (2): 878-889. https://doi.org/10.3906/mat-2101-93
