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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 (xm yn ) = F (yn xm ), (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.

Let R be an associative ring. A mapping f : R −→ R is said to be additive if f (x +y) = f (x) +f (y) holds for all x;y ∈ R: An additive mapping d : R −→ R is called a derivation if d(xy) = d(x)y + xd(y) holds for all x;y ∈ R: In this paper, we investigate commutativity of prime and semiprime rings satisfying certain identities involving additive mappings and derivations. Moreover, some results have also been discussed.

Let [Formula: see text] be an associative ring with center [Formula: see text] The objective of this paper is to discuss the commutativity of a semiprime ring [Formula: see text] which admits a derivation [Formula: see text] such that [Formula: see text] for all [Formula: see text] or [Formula: see text] for all [Formula: see text] or [Formula: see text] for all [Formula: see text] where [Formula: see text] and [Formula: see text] are fixed positive integers. Finally, we apply these purely ring theoretic results to obtain commutativity of Banach algebra via derivation.