ON DERIVATIONS SATISFYING CERTAIN IDENTITIES ON RINGS AND ALGEBRAS

Abstract

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 $G(xy)\pm [F(x),y]\pm [x,y]\in Z(R)$ and $G(xy)\pm [x,F(y)]\pm [x,y]\in Z(R)$ 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 $G((xy)^{n})+[F(x^{n}),y^{n}]+[x^{n},y^{n}]\in Z(A)$ or $G((xy)^{n})- [F(x^{n}),y^{n}]-[x^{n},y^{n}]\in Z(A)$.