On Strongly pi-Regular Rings with Involution

TL;DR

This paper explores strongly pi-star-regular rings, extending their properties and characterizations, and connects these concepts with involution, building on recent advances in the field.

Contribution

It introduces new properties and characterizations of strongly pi-star-regular rings with involution, extending previous results in the literature.

Findings

Established new properties of strongly pi-star-regular rings.

Provided several characterizations in terms of (strong) pi-regularity and involution.

Extended recent results on pi-star-regular and star-periodic rings.

Abstract

Recall that a ring R is called strongly pi-regular if, for every a in R, there is a positive integer n, depending on a, such that a^n belongs to the intersection of a^{n+1}R and Ra^{n+1}. In this paper we give a further study of the notion of a strongly pi-star-regular ring, which is the star-version of strongly pi-regular rings and which was originally introduced by Cui-Wang in J. Korean Math. Soc. (2015). We also establish various properties of these rings and give several new characterizations in terms of (strong) pi-regularity and involution. Our results also considerably extend recent ones in the subject due to Cui-Yin in Algebra Colloq. (2018) proved for pi-star-regular rings and due to Cui-Danchev in J. Algebra Appl. (2020) proved for star-periodic rings.