Turkish Journal of Mathematics
DOI
10.3906/mat-1902-105
Abstract
We completely characterize a subclass of Drazin inverses by means of tripotents and nilpotents. We prove that an element $a$ in a ring $R$ has Hirano inverse if and only if $a^2\in R$ has strongly Drazin inverse, if and only if $a-a^3$ is nilpotent. If $\frac{1}{2}\in R$, we prove that $a\in R$ has Hirano inverse if and only if there exists $p^3=p\in comm^2(a)$ such that $a-p\in N(R)$, if and only if there exist two idempotents $e,f\in comm^2(a)$ such that $a+e-f\in N(R)$. Multiplicative and additive results for this generalized inverse are thereby obtained.
Keywords
Drazin inverse, nilpotent, tripotent, multiplicative property, Jacobson's lemma
First Page
2049
Last Page
2057
Recommended Citation
CHEN, HUANYIN and ABDOLYOUSEFI, MARJAN SHEIBANI
(2019)
"On Hirano inverses in rings,"
Turkish Journal of Mathematics: Vol. 43:
No.
4, Article 19.
https://doi.org/10.3906/mat-1902-105
Available at:
https://journals.tubitak.gov.tr/math/vol43/iss4/19
