Related papers: New characterizations for core inverses in rings w…

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 $(xa)^{\ast}=xa$, $xa^{2}=a$…

In this paper we introduce a new generalized inverse in a ring -- one-sided $(b, c)$-inverse, derived as an extension of $(b, c)$-inverse. This inverse also generalizes one-sided inverse along an element, which was recently introduced by H.…

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…

An element of a group is \emph{reversible} if it is conjugate to its own inverse, and it is \emph{strongly reversible} if it is conjugate to its inverse by an involution. A group element is strongly reversible if and only if it can be…

In this paper, we introduce new representation and characterization of the weighted core inverse of matrices. Several properties of these inverses and their interconnections with other generalized inverses are also explored. Through…

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 $\varphi$…

Let $S$ be a $*$-monoid and let $a,b,c$ be elements of $S$. We say that $a$ is $(b,c)$-core-EP invertible if there exist some $x$ in $S$ and some nonnegative integer $k$ such that $cax(ca)^{k}c=(ca)^{k}c$, $x{\mathcal R}(ca)^{k}b$ and…

Using the recent notion of inverse along an element in a semigroup, and the natural partial order on idempotents, we study bicommuting generalized inverses and define a new inverse called natural inverse, that generalizes the Drazin inverse…

An element $g$ of a group is called {\em reversible} if it is conjugate in the group to its inverse. This paper is about reversibles in the group $G$ of formally-invertible pairs of formal power series in two variables, with complex…

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…

We study the Drazin inverses of the sum and product of two elements in a ring. For Drazin invertible elements $a$ and $b$ such that $a^2b=aba$ and $b^2a=bab$, it is shown that $ab$ is Drazin invertible and that $a+b$ is Drazin invertible if…

Let $R$ be a unital ring with involution.In this paper, several new necessary and sufficient conditions for the existence of the Moore-Penrose inverse of an element in a ring $R$ are given.In addition, the formulae of the Moore-Penrose…

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…

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…

The theory of generalized inverses of matrices and operators is closely connected with projections, i.e., idempotent (bounded) linear transformations. We show that a similar situation occurs in any associative ring $\mathcal{R}$ with a unit…

Let $\mathscr{C}$ be a category with an involution $\ast$. Suppose that $\varphi : X \rightarrow X$ is a morphism and $(\varphi_1, Z, \varphi_2)$ is an (epic, monic) factorization of $\varphi$ through $Z$, then $\varphi$ is core invertible…

We introduce and study a new class of Drazin inverses. An element $a$ in a ring $R$ has Drazin inverse $b$ if $a^2-ab\in N(R)$, $ab=ba$ and $b=bab$. Every Hirano inverse of an element is its Drazin inverse.We drive several characterization…

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…

An element $g$ of a group is called reversible if it is conjugate in the group to its inverse. An element is an involution if it is equal to its inverse. This paper is about factoring elements as products of reversibles in the group…

Let $\mathscr{C}$ be an additive category with an involution $\ast$. Suppose that $\varphi : X \rightarrow X$ is a morphism of $\mathscr{C}$ with core inverse $\varphi^{\co} : X \rightarrow X$ and $\eta : X \rightarrow X$ is a morphism of…