# Rendiconti del Seminario Matematico della Università di Padova

Volume 121, 2009, pp. 243–257
DOI: 10.4171/RSMUP/121-15

Published online: 2009-06-30

Right Sided Ideals and Multilinear Polynomials with Derivation on Prime Rings

Basudeb Dhara[1] and Rajendra K. Sharma[2]

(1) Belda College, Paschim Medinipur, India
(2) Indian Institute of Technology, New Delhi, India

Let R be an associative prime ring of char R ≠ 2 with center Z(R) and extended centroid C, f(x1, ...,xn) a nonzero multilinear polynomial over C in n noncommuting variables, d a nonzero derivation of R and ρ a nonzero right ideal of R. We prove that: (i) if [d2(f(x1, ...,xn)), f(x1, ...,xn)] = 0 for all x1, ...,xnρ then ρC = eRC for some idempotent element e in the socle of RC and f(x1, ...,xn) is central-valued in eRCe unless d is an inner derivation induced by bQ such that b2 = 0 and = 0; (ii) if [d2(f(x1, ...,xn)), f(x1, ...,xn)] ∈ Z(R) for all x1, ...,xnρ then ρC=eRC for some idempotent element e in the socle of RC and either f(x1, ...,xn) is central in eRCe or eRCe satisfies the standard identity S4(x1,x2, x3,x4) unless d is an inner derivation induced by bQ such that b2 = 0 and = 0.