On Some Extensions of π-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.