JORDAN u-GENERALIZED REVERSE DERIVATIONS ON SEMIPRIME RINGS

In this paper, we prove that if G is a Jordan u-generalized reverse derivation of a semi prime ring R of char.≠2, then G is a u-generalized reverse derivation. Similarly, we show that if G is a Jordan u−* generalized reverse derivation of a semi prime ring R of char.≠2, then G is a u−*generalized reverse derivation. We also prove that the commutativity of R if G((x,y)) =0. INTRODUCTION: M.Bresar (2) proved that for a semiprime ring R, if G is a function from R to R and D:R→ R is an additive mapping such that G(xy )=G(x)y+ xD (y), for all x,y∈R,then D is uniquely determined by G and moreover G must be a derivation. In (1) Ashraf and Rehman proved that if R is a ring of char.≠2 such that R has a commutator which is not a zero divisor, then every Jordan generalized derivation on R is a generalized derivation.We know that an additive mapping G:R→ R is a Jordan generalized reverse derivation if there exists a derivation D from R to R such that G(x 2 )=G(x)x+ xD (x), for all x in R. An additive mapping Dfrom R to itself is a u-reverse derivation if D(xy )= D(y)u(x)+ yD (x) hold where u is a homomorphism of R, for all x,yin R. An additive mapping G:R→ R is a u-generalized reverse derivation if there exists a derivation D from R to R such that G(xy )=G(y)u(x)+ yD (x),for all x,y∈R and an additive mapping G:R→ R is a Jordan u- generalized reverse derivation if there exists a derivation D:R→ R such that G(x 2 )=G(x)u(x)+ xD (x),for all x∈R. An additive mapping D from R to itself is a u−* reverse derivation if D(xy )=u(x)D(y)+D(x)y, where u is an anti - homomorphism in R, for all x,y∈R. An additive mapping G:R→ R is a u−* generalized reverse derivation if there exists a derivation D from R to R such that G(xy )=u(x)G(y)+D(x)y, where u is an anti-homomorphism in R, for all x,y∈Rand an additive mapping G:R→ R is a Jordan u−*generalized reverse derivation if there exists a derivation from R to R such that G(x 2 )= u(x)G(x)+D(x)x, where u is an anti-homomorphism in R, for all x∈R. Clearly, every u-generalized derivation on a ring is a Jordan u-generalized derivation. But the converse statement does not hold in general. Throughout this paper R will be a semiprime ring and Z its center.