Article
Kyungpook Mathematical Journal 2024; 64(1): 15-30
Published online March 31, 2024 https://doi.org/10.5666/KMJ.2024.64.1.15
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
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 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: EP element, Normal EP element, Strongly EP element, Partial isometry