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European Mathematical Society Publishing House
2016-09-19 17:05:48
Rendiconti del Seminario Matematico della Università di Padova
Rend. Sem. Mat. Univ. Padova
RSMUP
0041-8994
2240-2926
General
10.4171/RSMUP
http://www.ems-ph.org/doi/10.4171/RSMUP
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European Mathematical Society Publishing House
Zuerich, Switzerland
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121
2009
0
Right Sided Ideals and Multilinear Polynomials with Derivation on Prime Rings
Basudeb
Dhara
Belda College, PASCHIM MEDINIPUR, INDIA
Rajendra
Sharma
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, ...,xn ∈ ρ 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 b ∈ Q such that b2 = 0 and bρ = 0; (ii) if [d2(f(x1, ...,xn)), f(x1, ...,xn)] ∈ Z(R) for all x1, ...,xn ∈ ρ 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 b ∈ Q such that b2 = 0 and bρ = 0.
Associative rings and algebras
General
243
257
10.4171/RSMUP/121-15
http://www.ems-ph.org/doi/10.4171/RSMUP/121-15