A note on abelian stongly k-Engel π-regular rings

Let R be an associative ring with identity and let k ≥ 1 be a fixed integer. An element (x, y) ∈ R × R is said to be left (right) k-Engel π-regular if there exists a positive integer n and an element z ∈ R such that [ x, y]^n_k = z[ x, y]^n+1_k ([ x, y]^n_k =[ x, y]^n+1_k z). If every element of R × R is left (right) k-Engel π-regular, then R is said to be left (right) k- Engel π-regular. An element (x, y) ∈ R × R is strongly k-Engel π-regular if it is both left and right k-Engel π-regular. The ring R is strongly k-Engel π-regular if every element of R × R is strongly k-Engel π-regular. In this paper, we investigate properties of abelian strongly k-Engel π-regular ring.