On Some Extensions of $\pi$-Regular Rings

Some variations of $\pi$-regular and nil clean rings were recently introduced in \cite{5,8,7}, respectively. 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 \cite{8}. Moreover, some other analogous questions concerning the symmetric properties of certain classes of rings are treated as well by proving that centrally Utumi rings are always strongly $\pi$-regular.