  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.
