New characterizations for core inverses in rings with involution

Abstract

The core inverse for a complex matrix was introduced by O. M. Baksalary and G. Trenkler. D. S. Rakíc, N.Č. Diňcíc and D. S. Djordjevíc 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, ba²= a, and ab²= 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.

Keywords

Core inverse / dual core inverse / group inverse / {1,3}-inverse / {1,4}-inverse

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Higher Education Press and Springer-Verlag Berlin Heidelberg