Abstract
The core inverse for a complex matrix was introduced by O. M. Baksalary and G. Trenkler. D. S. Rakić, N. Č. Dinčić and D. S. Djordjević 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.
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Xu, S., Chen, J. & Zhang, X. New characterizations for core inverses in rings with involution. Front. Math. China 12, 231–246 (2017). https://doi.org/10.1007/s11464-016-0591-2
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DOI: https://doi.org/10.1007/s11464-016-0591-2