New characterizations for core inverses in rings with involution

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• 作者：Sanzhang Xu ; Jianlong Chen ; Xiaoxiang Zhang
• 关键词：Core inverse ; dual core inverse ; group inverse ; {1 ; 3} ; inverse ; {1 ; 4} ; inverse
• 刊名：Frontiers of Mathematics in China
• 出版年：2017
• 出版时间：February 2017
• 年：2017
• 卷：12
• 期：1
• 页码：231-246
• 全文大小：170 KB
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Mathematics
  Chinese Library of Science
  
• 出版者：Higher Education Press, co-published with Springer-Verlag GmbH
• ISSN：1673-3576
• 卷排序：12

文摘

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, 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.