New characterizations for core inverses in rings with involution

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 inverse of an element in a ring can be characterized by five equations and every core invertible element is group invertible. It is natural to ask when a group invertible element is core invertible, in this paper, we will answer this question. We will use three equations to characterize the core inverse of an element. That is, let $a,b\in R$, then $a\in R^{\tiny\textcircled{\tiny#}}$with$a^{\tiny\textcircled{\tiny#}}=b$ if and only if $(ab{)}^{\ast }=ab$, $b{a}^2=a$ and $a{b}^2=b$. Finally, we investigate the additive property of two core invertible elements. Moreover, the formulae of the sum of two core invertible elements are presented.