Sandhu, Gurninder S.Kumar, DeepakCamci, Didem K.Aydin, Neset2025-01-272025-01-2720190352-96652406-047Xhttps://doi.org/10.22190/FUMI1901085Shttps://hdl.handle.net/20.500.12428/24778The present paper deals with the commutativity of an associative ring R and a unital Banach Algebra A via derivations. Precisely, the study of multiplicative (generalized)-derivations F and G of semiprime (prime) ring R satisfying the identities G(xy) +/- [F(x), y] +/- [x, y] is an element of Z(R) and G(xy) +/- [x , F(y)] +/- [x, y] is an element of Z(R) has been carried out. Moreover, we prove that a unital prime Banach algebra A admitting continuous linear generalized derivations F and G is commutative if for any integer n > 1 either G((xy)(n)) + [F(x(n)), y(n) ] + [x(n),y(n)] is an element of Z(A) or G((xy(n)) - [F(x(n)), y(n)] - [x(n) , y(n)] is an element of Z(A).eninfo:eu-repo/semantics/closedAccessBanach algebraAssociative ringGeneralized derivationsArticle341859910.22190/FUMI1901085SN/AWOS:000461026100008