摘要
The core inverse for a complex matrix was introduced by O. M. Baksalary and G. Trenkler. D. S. Rakic, N. C. Dincic and D. S. Djordjevc 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. Let R be a ring with involution, we will use three equations to characterize the core inverse of an element. That is, let a,b ∈ R. Then a ∈ R with a= b if and only if （ab）^＊ = 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.
The core inverse for a complex matrix was introduced by O. M. Baksalary and G. Trenkler. D. S. Rakic, N. C. Dincic and D. S. Djordjevc 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. Let R be a ring with involution, we will use three equations to characterize the core inverse of an element. That is, let a,b ∈ R. Then a ∈ R with a= b if and only if （ab）^＊ = 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.
基金
Acknowledgements This work was supported by the National Natural Science Foundation of China （Grant Nos. 11201063, 11371089）, the Specialized Research Fund for the Doctoral Program of Higher Education （No. 20120092110020）, the Jiangsu Planned Projects for Postdoctoral Research Funds （No. 1501048B）, and the Natural Science Foundation of Jiangsu Province （No. BK20141327）.
