**Power Values of Derivations on Multilinear Polynomials in Prime Rings **

* Basudeb Dhara, Sukhendu Kar, Sachhidananda Mondal *

**Abstract**

Let *R* be a prime ring with center *Z*(*R*) and with extended centroid *C*, *d* a derivation of *R* and *f*(*x* _{1},…,*x* _{ n }) a nonzero multilinear polynomial over *C*, *m* ≥ 1 and *p* ≥ 1 two integers. In the present paper, we study the situations (i) ((*d*(*f*(*x* _{1},…,*x* _{ n })))^{ m } − *f*(*x* _{1},…,*x* _{ n }))^{ p } = 0; (ii) ((*d*(*f*(*x* _{1},…,*x* _{ n })))^{ m } − *f*(*x* _{1},…,*x* _{ n }))^{ p } ∈ *Z*(*R*) for all *x* _{1},…,*x* _{ n } in some subsets of *R*.