On rings and algebras with derivations

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.