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Left w-Core Inverses in Rings with Involution

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Abstract

In Zhu et al. (Linear Multilinear Algebra 71:528–544, 2023), the authors described the left w-core inverse by principal ideals in \(*\)-ring, and asked whether it can be defined by the solution of equations. In this paper, we answer the question in the positive. For any \(*\)-ring R and \(a,w\in R\), the element a is called left w-core invertible if there is some \(x\in R\) satisfying \(awxa=a\), \(xawa=a\) and \((awx)^{*}=awx\). Several criteria for left w-core inverses are presented. Among of these, it is proved that a is left w-core invertible if and only if w is left invertible along a, a (or aw) is \(\{1,3\}\)-invertible and \(a\in awR\). Also, the relations among left w-core inverses, w-core inverses, and other generalized inverses are established. As applications, several characterizations for the Moore–Penrose inverse, the core inverse, and the pseudo-core inverse are given.

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Acknowledgements

The authors are grateful to the referees for their careful reading and suggesting comments which led to improvement of this paper. This research is supported by the National Natural Science Foundation of China (No. 11801124 and No. 12371023) and China Postdoctoral Science Foundation (No. 2020M671068).

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Authors and Affiliations

  1. School of Mathematics, Hefei University of Technology, Hefei, 230009, China

    Huihui Zhu & Chengcheng Wang

  2. Department of Mathematics and Newtouch Center for Mathematics, Shanghai University, Shanghai, 200444, China

    Qing-Wen Wang

  3. Collaborative Innovation Center for the Marine Artificial Intelligence, Shanghai University, Shanghai, 200444, China

    Qing-Wen Wang

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  1. Huihui Zhu
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  2. Chengcheng Wang
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  3. Qing-Wen Wang
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Correspondence to Qing-Wen Wang.

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Zhu, H., Wang, C. & Wang, QW. Left w-Core Inverses in Rings with Involution. Mediterr. J. Math. 20, 337 (2023). https://doi.org/10.1007/s00009-023-02541-9

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  • DOI: https://doi.org/10.1007/s00009-023-02541-9

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