The further results on SEP elements in a ring with involution

Year 2026, Volume: 39 Issue: 39, 121 - 142, 10.01.2026

Xinran Wang Yan Ji Junchao Wei

https://doi.org/10.24330/ieja.1764204

https://izlik.org/JA97BT87XR

Abstract

In this paper, we further study many new characterizations of $SEP$ elements in a ring with involution. Firstly, combining Moore-Penrose invertible element, group invertible element, we find some $PE$ elements to characterize $SEP$ elements and then further discover some equivalent conditions for $SEP$ elements especially around the element $aa^*a^+a$. Mainly, by constructing some equations in a given set including $a^+, a^*, (a^\#)^*, a^+a, aa^+$, we obtain a lot of new characterizations of $SEP$ elements. Next, we study the expression forms of related bivariate equations to depict $SEP$ elements. Finally, we use nil-cleanity of the element $aa^*a^+a$ to link $SEP$ elements with $PE$ elements.

Keywords

$SEP$ element , $PE$ element , $EP$ element , solution of equation , bivariate equation

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