JournalsrsmupVol. 121falsepp. 243–257

Right Sided Ideals and Multilinear Polynomials with Derivation on Prime Rings

  • Basudeb Dhara

    Belda College, Paschim Medinipur, India
  • Rajendra K. Sharma

    Indian Institute of Technology, New Delhi, India
Right Sided Ideals and Multilinear Polynomials with Derivation on Prime Rings cover

Abstract

Let R be an associative prime ring of char R ≠ 2 with center Z(R) and extended centroid C, f(_x_1, ...,_x_n) 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 [_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 idempotent element e in the socle of RC and f(_x_1, ...,_x_n) is central-valued in eRCe unless d is an inner derivation induced by bQ such that _b_2 = 0 and = 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 idempotent element e in the socle of RC and either f(_x_1, ...,_x_n) is central in eRCe or eRCe satisfies the standard identity _S_4(_x_1,_x_2, _x_3,_x_4) unless d is an inner derivation induced by bQ such that _b_2 = 0 and = 0.

Cite this article

Basudeb Dhara, Rajendra K. Sharma, Right Sided Ideals and Multilinear Polynomials with Derivation on Prime Rings. Rend. Sem. Mat. Univ. Padova 121 (2009), pp. 243–257

DOI 10.4171/RSMUP/121-15