Related papers: On Hirano inverses in rings

Let $R$ be a ring with identity and $J(R)$ be its Jacobson radical. Assume that $a\in R$ is $(b,c)$-invertible and $j_a,j_b,j_c\in J(R)$. This paper provides necessary and sufficient conditions for $a+j_a$ to be $(b+j_b,c+j_c)$-invertible.…

We present new generalized Jacobson's lemma for generalized Drazin inverses. This extend the main results on g-Drazin inverse of Yan, Zeng and Zhu (Linear $\&$ Multilinear Algebra, {\bf 68}(2020), 81--93).

We apply recent results on the rank of elements of rings to study the structure of generalized corner rings $aRa$, where $R$ is a unital ring and $a$ an element of $R$. We give a complete description of the structure of $aRa$ when $a^2$ has…

An element of a group is said to be reversible if it is conjugate to its inverse. We characterise the reversible elements in the group of diffeomorphisms of the real line, and in the subgroup of order preserving diffeomorphisms.

In this paper, we first prove that if a is both left (b, c)-invertible and left (c, b)-invertible, then a is both (b, c)-invertible and (c, b)-invertible in a *-monoid, which generalized the recent result about the inverse along an element…

In this paper, some Drazin inverse representations of the linear combinations of two idempotents in Banach algebra are obtained.

In this paper we give expressions for the generalized Drazin inverse of a (2,2,0) operator matrix and a $2\times2$ operator matrix under certain circumstances, which generalizes and unifies several results in the literature.

The aim of this paper is to introduce and study left and right versions of the class of generalized Drazin-Riesz invertible operators, as well as left and right versions of the class of generalized Drazin-meromorphic invertible operators.

We define here the notion of a {\it weakly reversible ring} $R$ saying that a non-zero element $a\in R$ is weakly reversible if there exists an integer $m>0$ depending on $a$ such that $a^m\neq 0$ is reversible, that is,…

The Drazin index is a fundamental invariant in the analysis of singular matrices and their generalized inverses. While sharp results are available for block triangular matrices, the corresponding theory for anti-triangular block matrices is…

For $V$ a vector space over a field, or more generally, over a division ring, it is well-known that every $x\in\mathrm{End}(V)$ has an <i>inner inverse</i>, i.e., an element $y\in\mathrm{End}(V)$ satisfying $xyx=x.$ We show here that a…

Let $R$ be a ring with involution containing a nontrivial symmetric idempotent element $e$. Let $\delta: R\rightarrow R$ be a mapping such that $\delta(ab)=\delta(b)a^{\ast}+b^{\ast}\delta(a)$ for all $a,b\in R$, we call $\delta$ a…

Let $R$ be a B\'ezout domain, and let $A,B,C\in R^{n\times n}$ with $ABA=ACA$. If $AB$ and $CA$ are group invertible, we prove that $AB$ is similar to $CA$. Moreover, we have $(AB)^{\#}$ is similar to $(CA)^{\#}$. This generalize the main…

In this paper, we study B-Fredholm elements in rings and algebras. After characterising these elements in terms of generalized Fredholm elements, we will give a condition on the socle of a unital primitive Banach algebra $A$, under which we…

We consider the problem of classifying (possibly noncommutative) R-algebras of low rank over an arbitrary base ring R. We first classify algebras by their degree, and we relate the class of algebras of degree 2 to algebras with a standard…

Given a Banach Algebra $A$ and $a\in A$, several relations among the Drazin spectrum of $a$ and the Drazin spectra of the multiplication operators $L_a$ and $R_a$ will be stated. The Banach space operator case will be also examined.…

In this paper we present expressions for the Drazin inverse of the generalized Schur complement $A-CD^{d}B$ in terms of the Drazin inverses of $A$ and the generalized Schur complement $D-BA^{d}C$ under less and weaker restrictions, which…

In this paper, we discuss the common properties for the products $ac$ and $ba$ in various categories under the condition $a(ba)^{2}=abaca=acaba=(ac)^{2}a$. We prove that generalized Jacobson's lemma and Cline's formula are suitable for…

We characterize the generalized weighted core-EP inverse via the canonical decomposition, utilizing a weighted core-EP invertible element and a quasinilpotent. We then offer a polar-like characterization for the generalized weighted core-EP…

Let $\mathcal{R}$ be a unital ring with involution. The notions of 1MP-inverse and MP1-inverse are extended from $M_{m,n}(\mathbb{C)}$, the set of all $m\times n $ matrices over $\mathbb{C}$, to the set $\mathcal{R}% ^{\dagger}$ of all…