  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.
