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OPEN ACCESS eISSN 0454-8124
pISSN 1225-6951
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Kyungpook Mathematical Journal 2024; 64(1): 15-30

Published online March 31, 2024

Copyright © Kyungpook Mathematical Journal.

Generalized Inverses and Solutions to Equations in Rings with Involution

Yue Sui and Junchao Wei

Department of Mathematics, Yangzhou University,Yangzhou, 225002, P. R. China
e-mail : suiyue052@126.com and jcweiyz@126.com

Received: May 4, 2021; Revised: June 23, 2022; Accepted: July 25, 2022

Abstract

In this paper, we focus on partial isometry elements and strongly EP elements or 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 aR#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 aR#R is a strongly EP element if and only if the equation (a)*xa = xaa has at least one solution in {a, a#, a, a*, (a#)*, (a)*}. These characterizations extend many well-known results.

Keywords: EP element, Normal EP element, Strongly EP element, Partial isometry