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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 greater than or equal to 1, then a is core invertible if and only if there exists a Hermitian element (or a projection) p such that pa=0, a^n+p is invertible. As a consequence, a is an EP element if and only if there exists a Hermitian element (or a projection) p such that pa=ap=0, a^n+p is invertible. We also get a new characterization for both core invertible and dual core invertible of a regular element by units, and their expressions are shown. In particular, we prove that for n is an integer greater than or equal to 2, a is both Moore-Penrose invertible and group invertible if and only if (a*)^n is invertible along 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 $ain R$ and $ein R$ be an invertible Hermitian element, $ngeqslant
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 $varphi$ has a cokernel $lam
In this paper, we introduce two new generalized inverses of matrices, namely, the $bra{i}{m}$-core inverse and the $pare{j}{m}$-core inverse. The $bra{i}{m}$-core inverse of a complex matrix extends the notions of the core inverse defined by Baksalar
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 $mathscr{C}$ such tha
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 if and only if $(v