Commutativity of rings involving additive mappings

Abstract

Let R be an associative ring. In the present paper, we investigate commutativity of a ring admitting an additive mapping F satisfying any one of the following properties: (i) F ([x, y]² ) = F ([x² , y² ]), (ii) F ((x ◦ y)² ) = F (x² ◦ y² ), (iii) F ((xy)ⁿ ) = F (xⁿ yⁿ ), (iv) F (x^m yⁿ ) = F (yⁿ x^m ), (v) (F (x)F (y))ⁿ = (F (y)F (x))ⁿ for all x, y ∈ R, where m and n are positive integers greater than 1. Moreover, some related results are also discussed. Finally, some examples are given to demonstrate that the restrictions imposed on the hypotheses of the various results are not superfluous.