ON DERIVATIONS SATISFYING CERTAIN IDENTITIES ON RINGS AND ALGEBRAS

The 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).