arXiv subjects

Honglin Zou

Publications and source records attributed to Honglin Zou.

3 recordsLinked to original sources

arXiv2026

On $\ast$-Reversible and Generalized $\ast$-Reversible Rings

Let $R$ be a $\ast$-ring with $a,b\in R$. A ring $R$ is said to be $\ast$-reversible if $ab=0$ implies $b^{\ast}a=0$. In this paper, we first establish several new characterizations of $\ast$-reversible rings and reversible rings. In particular, we prove that $\ast$-reversible rings coincide with $\ast$-symmetric rings. Using these characterizations, we introduce two new classes of generalized $\ast$-reversible rings: pro-$\ast$-reversible rings and nil-$\ast$-reversible rings. A ring $R$ is called pro-$\ast$-reversible if $ab\in P(R)$ implies $b^{\ast}a\in P(R)$, and $R$ is nil-$\ast$-reversible if for every $c\in N(R)$, $cb=0$ yields both $b^{\ast}c=0$ and $cb^{\ast}=0$. The basic properties and characterizations of pro-$\ast$-reversible and nil-$\ast$-reversible rings are investigated. The interrelationships among all these ring classes are considered. The related examples to distinguish these rings are constructed.

math.RA↗

arXiv2026

Transposed Triple Products and Pro-Symmetric Rings in $\ast$-Rings

This paper investigates properties concerning transposed triple products in rings. Motivated by Cline's formula, we characterize symmetric rings by means of group invertible elements and EP elements. We prove that a unital ring $R$ is symmetric if and only if $abc\in R^{\sharp}$ implies $acb\in R^{\sharp}$ for all $a,b,c\in R$. In particular, we give an answer to the problem posed in \cite[Problem 2.9]{MW1}. An example is provided to illustrate that for a symmetric ring $R$, $abc=e$ does not generally yield $acb=e$. For $\ast$-rings, we introduce the notion of pro-symmetric rings: a ring $R$ is pro-symmetric if $abc\in P(R)$ implies $acb\in P(R)$ for all $a,b,c\in R$. We show that every pro-symmetric ring is symmetric. Several counterexamples are constructed to distinguish these classes of rings, and their mutual inclusion relations are also discussed.

math.RA↗

arXiv2023

On the g$π$-Hirano invertibility in Banach algebras

In a Banach algebra, we introduce a new type of generalized inverse called g$π$-Hirano inverse. Firstly, several existence criteria and the equivalent definition of this inverse are investigated. Then, we discuss the relationship between the g$π$-Hirano invertibility of $a$, $b$ and that of the sum $a+b$ under some weaker conditions. Finally, as applications to the previous additive results, some equivalent characterizations for the g$π$-Hirano invertibility of the anti-triangular matrix over Banach algebras are obtained.In particular, some results in this paper are different from the corresponding ones of classical generalized inverses, such as Drazin inverse and generalized Drazin inverse.

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