Journal of Algebra and Its Applications Vol. 15, No. 6 (2016) 1650107 (10 pages) c  World Scientific Publishing Company DOI: 10.1142/S0219498816501073 On rings and algebras with derivations Shakir Ali ∗ Department of Mathematics, Faculty of Science – Rabigh King Abdulaziz University Jeddah-21589, Saudi Arabia sashah@kau.edu.sa shakir50@rediffmail.com Mohammad Salahuddin Khan † and Abdul Nadim Khan ‡ Department of Mathematics Aligarh Muslim University Aligarh-202002, India † salahuddinkhan50@gmail.com ‡ abdulnadimkhan@gmail.com Najat M. Muthana Department of Mathematics Faculty of Science for Girls King Abdulaziz University Jeddah-21589, Saudi Arabia nmuthana@kau.edu.sa Received 16 January 2014 Accepted 22 June 2015 Published 2 September 2015 Communicated by C. M. Ringel 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 ] ∈ Z(R) for all x, y ∈ R or d([x m ,y n ]) ∈ Z(R) for all x, y ∈ R or d(x m ◦ y n ) ∈ Z(R) for all x, y ∈ 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. Keywords : Semiprime ring; Banach algebra; derivation. Mathematics Subject Classification: 16W25, 16N60, 16U80, 46J45 1. Introduction This research is motivated by the work of Bell [5] and Yood [34]. Throughout this paper R will denote an associative ring with center Z (R), and A will represent an ∗ Permanent address: Department of Mathematics, Aligarh Muslim University, Aligarh-202002, India. 1650107-1 J. Algebra Appl. Downloaded from www.worldscientific.com by Dr. Mohammad Salahuddin Khan on 09/07/15. For personal use only.