Characterizations of weighted core inverse in rings with involution

$R$ 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, $ngeqslant 1$, then $a$ is $e$-core invertible if and only if there exists an element (or an idempotent) $p$ such that $(ep)^{ast}=ep$, $pa=0$ and $a^{n}+p$ (or $a^{n}(1-p)+p$) is invertible. As a consequence, let $e, fin R$ be two invertible Hermitian elements, then $a$ is weighted-$mathrm{EP}$ with respect to $(e, f)$ if and only if there exists an element (or an idempotent) $p$ such that $(ep)^{ast}=ep$, $(fp)^{ast}=fp$, $pa=ap=0$ and $a^{n}+p$ (or $a^{n}(1-p)+p$) is invertible. These results generalize and improve conclusions in cite{Li}.