Related papers: A class of core inverses associated with Green's r…

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…

Let $a,b,c\in R$ where $R$ is a $*$-ring. We call $a$ \textit{left dual $(b,c)$-core invertible} if there exists $x\in Rc$ such that $bxab=b$ and $(xab)^*=xab$. Such an $x$ is called a left dual $(b,c)$-core inverse of $a$. In this paper,…

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…

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…

Let $R$ be a unital $*$-ring. For any $a,w,b\in R$, we apply the defined $w$-core inverse to define a new class of partial orders in $R$, called the $w$-core partial order. Suppose $a,b\in R$ are $w$-core invertible. We say that $a$ is…

Specific definitions of the core and core-EP inverses of complex tensors are introduced. Some characterizations, representations and properties of the core and core-EP inverses are investigated. The results are verified using specific…

Existence criteria for the $(b,c)$-inverse are given.% in terms of annihilators. We present explicit expressions for the $(b,c)$-inverse by using inner inverses. We answer the question when the $(b,c)$-inverse of $a\in R$ is an inner…

In this paper, we introduce the notation of $E$-weighted core-EP and $F$-weighted dual core-EP inverse of matrices. We then obtain a few explicit expressions for the weighted core-EP inverse of matrices through other generalized inverses.…

We present a necessary and sufficient conditions under which the sum of two EP elements in a *-ring has core inverse. As an application, we establish the conditions under which a block complex matrix with EP sub-blocks has core inverse.

We study generalized inverses for matrices associated with double star digraphs. Explicit block formulas and existence criteria are obtained for core, dual core, core EP, and dual core EP inverses, expressed in terms of explicit algebraic…

Let $R$ be a unital ring with involution. We first show that the EP elements in $R$ can be characterized by three equations. Namely, let $a\in R$, then $a$ is EP if and only if there exists $x\in R$ such that $(xa)^{\ast}=xa$, $xa^{2}=a$…

$R$ is a unital ring with involution. We investigate the characterizations and representations of weighted core inverse of an element in $R$ by idempotents and units. For example, let $a\in R$ and $e\in R$ be an invertible Hermitian…

The core inverse for a complex matrix was introduced by Baksalary and Trenkler. Raki\'c, Din\v{c}i\'c and Djordjevi\'c generalized the core inverse of a complex matrix to the case of an element in a ring. They also proved that the core…

Let $\mathscr{C}$ be an additive category with an involution $\ast$. Suppose that $\varphi : X \rightarrow X$ is a morphism with kernel $\kappa : K \rightarrow X$ in $\mathscr{C}$, then $\varphi$ is core invertible if and only if $\varphi$…

In this paper, firstly we study the continuity of the core-EP inverse without explicit error bounds by virtue of two methods. One is the rank equality, followed from the classical generalized inverse. The other one is matrix decomposition.…

Let R be a unital ring with involution, we give the characterizations and representations of the core and dual core inverses of an element in R by Hermitian elements (or projections) and units. For example, let a in R and n is an integer…

In this paper, we introduce the notion of a (generalized) right core inverse and give its characterizations and expressions. Then, we provide the relation schema of (one-sided) core inverses, (one-sided) pseudo core inverses and EP…

The notion of weighted $(b,c)$-inverse of an element in rings were introduced, very recently [Comm. Algebra, 48 (4) (2020): 1423-1438]. In this paper, we further elaborate on this theory by establishing a few characterizations of this…

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…

In this article the $(b, c)$-inverse will be studied. Several equivalent conditions for the existence of the $(b,c)$-inverse in rings will be given. In particular, the conditions ensuring the existence of the $(b,c)$-inverse, of the…