Right Sided Ideals and Multilinear Polynomials with Derivation on Prime Rings

Abstract

Let R be an associative prime ring of char R ≠ 2 with centerZ(R) and extended centroid C, f(_x_1, ...,_x_n) a nonzeromultilinear polynomial over C in n noncommuting variables, da nonzero derivation of R and ρ a nonzero right ideal ofR. We prove that: (i) if[_d_2(f(_x_1, ...,_x_n)), f(_x_1, ...,_x_n)] = 0 for all _x_1, ...,_x_n ∈ ρ then ρ__C = eRC for some idempotentelement e in the socle of RC and f(_x_1, ...,_x_n) iscentral-valued in eRCe unless d is an inner derivation inducedby b ∈ Q such that _b_2 = 0 and bρ = 0; (ii) if[_d_2(f(_x_1, ...,_x_n)), f(_x_1, ...,_x_n)] ∈ Z(R) for all_x_1, ...,_x_n ∈ ρ then ρ__C=eRC for some idempotentelement e in the socle of RC and either f(_x_1, ...,_x_n) iscentral in eRCe or eRCe satisfies the standard identity_S_4(_x_1,_x_2,_x_3,_x_4) unless d is an inner derivation inducedby b ∈ Q such that _b_2 = 0 and bρ = 0.