On commutativity of rings and Banach algebras with generalized derivations

In the present paper, it is shown that if a prime ring R admits a generalized derivation f associated with a nonzero derivation d such that either f([x^{m},y^{n}])+[x^{m},y^{n}]\in Z(R)\quad\text{for all }x,y\in R or f([x^{m},y^{n}])-[x^{m},y^{n}]\in Z(R)\quad\text{for all }x,y\in R, then R is commutative. We apply this purely ring theoretic result to obtain commutativity of Banach algebras and prove that if A is a prime Banach algebra which admits a continuous linear generalized derivation f associated with a nonzero continuous linear derivation d such that either {f([x^{m},y^{n}])-[x^{m},y^{n}]\in Z(A)} or {f([x^{m},y^{n}])+[x^{m},y^{n}]\in Z(A)} for an integer {m=m(x,y)>1} and sufficiently many {x,y} in A, then A is commutative. A similar result is obtained for a unital prime Banach algebra A which admits a nonzero continuous linear generalized derivation f associated with a continuous linear derivation d such that {d(Z(A))\neq 0} satisfying either {f((xy)^{m})-x^{m}y^{m}\in Z(A)} or {f((xy)^{m})-y^{m}x^{m}\in Z(A)} for each {x\in G_{1}} and {y\in G_{2}} , where {G_{1},G_{2}} are open sets in A and {m=m(x,y)>1} is an integer: then A is commutative.