Kyungpook Mathematical Journal

Department of Mathematics, Kyungpook National University (경북대학교 자연과학대학 수학과)
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Generalized Inverses and Solutions to Equations in Rings with Involution

  • Yue Sui (Department of Mathematics, Yangzhou University) ;
  • Junchao Wei (Department of Mathematics, Yangzhou University)
  • Received : 2021.05.04
  • Accepted : 2022.07.25
  • Published : 2024.03.31
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Abstract

In this paper, we focus on partial isometry elements and strongly EP elements on a ring. We construct characterizing equations such that an element which is both group invertible and MP-invertible, is a partial isometry element, or is strongly EP, exactly when these equations have a solution in a given set. In particular, an element a ∈ R# ∩ R† is a partial isometry element if and only if the equation x = x(a†)*a† has at least one solution in {a, a#, a†, a*, (a#)*, (a†)*}. An element a ∈ R#∩R† is a strongly EP element if and only if the equation (a†)*xa† = xa†a has at least one solution in {a, a#, a†, a*, (a#)*, (a†)*}. These characterizations extend many well-known results.

Keywords

Acknowledgement

This work was supported by Jiangsu Students' Innovation and Entrepreneurship Training Program [202011117097Y] and the National Nature Science Foundation of China [11471282].

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