Abstract
Let R be an associative ring with center Z(R). The objective of this paper is to discuss the commutativity of a semiprime ring R which admits a derivation d such that d([x(m), y(n)]) +/- [x(m), y(n)] is an element of Z(R) for all x, y is an element of R or d([x(m), y(n)]) is an element of Z(R) for all x, y is an element of R or d(x(m) circle y(n)) is an element of Z(R) for all x, y is an element of R, where m and n are fixed positive integers. Finally, we apply these purely ring theoretic results to obtain commutativity of Banach algebra via derivation.