Turkish Journal of Mathematics
DOI
10.3906/mat-1903-92
Abstract
An element $a$ in a ring $R$ has a ps-Drazin inverse if there exists $b\in comm^2(a)$ such that $b=bab, (a-ab)^k\in J(R)$ for some $k\in {\Bbb N}$. Elementary properties of ps-Drazin inverses in a ring are investigated here. We prove that $a\in R$ has a ps-Drazin inverse if and only if $a$ has a generalized Drazin inverse and $(a-a^2)^k\in J(R)$ for some $k\in {\Bbb N}$. We show Cline's formula and Jacobson's lemma for ps-Drazin inverses. The additive properties of ps-Drazin inverses in a Banach algebra are obtained. Moreover, we completely determine when a $2\times 2$ matrix $A\in M_2(R)$ over a local ring $R$ has a ps-Drazin inverse.
Keywords
Generalized Drazin inverse, Cline's formula, Jacobson's lemma, $2\times 2$ matrix, local ring
First Page
2114
Last Page
2124
Recommended Citation
CHEN, HUANYIN and ÇALCI, TUĞÇE
(2019)
"On ps-Drazin inverses in a ring,"
Turkish Journal of Mathematics: Vol. 43:
No.
5, Article 3.
https://doi.org/10.3906/mat-1903-92
Available at:
https://journals.tubitak.gov.tr/math/vol43/iss5/3
