On some extensions of $\pi$-regular rings

Some variations of $\pi$-regular and nil clean rings were recently introduced in the works of the first author: "A generalization of $\pi$-regular rings, Turkish J. Math. 43 (2), 702–711 (2019)", "A symmetrization in $\pi$-regular rings, Trans. A. Razmadze Math. Inst. 174 (3), 271–275 (2020)", "A symmetric generalization of $\pi$-regular rings, Ric. Mat. 73 (1), 179–190 (2024)". In this paper, we examine the structure and relationships between these classes of rings. Specifically, we prove that $(m, n)$-regularly nil clean rings are left-right symmetric and also show that the inclusions ($D$-regularly nil clean) $\subseteq$ (regularly nil clean) $\subseteq$ ($(m,n)$-regularly nil clean) hold, as well as we answer Questions 1, 2 and 3 posed in the third of the above-listed works. Moreover, we also consider other similar questions concerning the symmetric properties of certain classes of rings. For example, it is proven that centrally Utumi rings are always strongly \(\pi\)-regular.