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 $(ab)^{\ast}=ab$, $ba^{2}=a$ and $ab^{2}=b$. Finally, we investigate the additive property of two core invertible elements. Moreover, the formulae of the sum of two core invertible elements are presented.