Related papers: Generalized inverses, ideals, and projectors in ri…

We consider generalized inverses of linear operators on arbitrary vector spaces and study the question when their product in reverse order is again a generalized inverse. This problem is equivalent to the question when the product of two…

A new generalized matrix inverse is derived which is consistent with respect to arbitrary nonsingular diagonal transformations, e.g., it preserves units associated with variables under state space transformations, thus providing a general…

Given an automorphism and an anti-automorphism of a semigroup of a Geometric Algebra, then for each element of the semigroup a (generalized) projection operator exists that is defined on the entire Geometric Algebra. A single fundamental…

Let $S$ be a $*$-semigroup and let $a,w,v\in S$. The initial goal of this work is to introduce two new classes of generalized inverses, called the $w$-core inverse and the dual $v$-core inverse in $S$. An element $a\in S$ is $w$-core…

We investigate generalized inverses of matrices associated with two classes of digraphs: double star digraphs and D-linked stars digraphs. For double star digraphs, we determine the Drazin index and derive explicit formulas for the Drazin…

Let $R$ be a ring with involution. The recently introduced notions of the core and dual core inverse are extended from matrix to an arbitrary $*$-ring case. It is shown that the group, Moore-Penrose, core and dual core inverse are closely…

Motivated by the very recent work of Gao, Y., Chen, J., Wang, J., Zou, H. [Comm. Algebra, 49(8) (2021) 3241-3254; MR4283143], we introduce two new generalized inverses named weak Drazin (WD) and weak Drazin Moore-Penrose (WDMP) inverses for…

A ring $R$ with an involution $*$ is a generalized Rickart $*$-ring if for all $x\in R$ the right annihilator of $x^n$ is generated by a projection for some positive integer $n$ depending on $x$. In this work, we introduce generalized right…

Generalized inverses of tensors play increasingly important roles in computational mathematics and numerical analysis. It is appropriate to develop the theory of generalized inverses of tensors within the algebraic structure of a ring. In…

Drazin inverses are a special kind of generalized inverses that can be defined for endomorphisms in any category. A natural question to ask is whether one can somehow extend the notion of Drazin inverse to arbitrary maps - not simply…

We give a general theory of generalised inverses and we explain the link with the theory of finitely generated projective modules. All the paper is written in constrctive mathematics in Bishop style. So all results do have a clear…

We study a ring containing a complete set of orthogonal idempotents as a generalized matrix ring via its Peirce decomposition. We focus on the case where some of the underlying bimodule homomorphisms are zero. Upper and lower triangular…

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.…

In this paper, we introduce the notion of weak core and central weak core inverse in a {\it proper $*$-ring}. We further elaborate on these two classes by producing a few representations and characterizations of the weak core and central…

We investigate properties of the group inverse in rings with unit related to products and differences of idempotents, and thus we extend some results from \cite{DENG} to more general settings. We show that most part of \cite{DENG} is easily…

Let $R$ be a ring with involution. In this paper, we introduce a new type of generalized inverse called pseudo core inverse in $R$. The notion of core inverse was introduced by Baksalary and Trenkler for matrices of index 1 in 2010 and then…

In this article several properties of the inverse along an element will be studied in the context of unitary rings. New characterizations of the existence of this inverse will be proved. Moreover, the set of all invertible elements along a…

The classes of 1MP-inverses and MP1-inverses are recently introduced classes of generalized inverses of complex matrix. Actually, they coincide with the classes of $\{1,2,3\}$ and $\{1,2,4\}$ inverses, respectively. We consider these…

Inversion of operators is a fundamental concept in data processing. Inversion of linear operators is well studied, supported by established theory. When an inverse either does not exist or is not unique, generalized inverses are used. Most…

We study generalized inverses on semigroups by means of Green's relations. We first define the notion of inverse along an element and study its properties. Then we show that the classical generalized inverses (group inverse, Drazin inverse…