Annihilator Conditions of Derivations on Multilinear Polynomials

Abstract

Let R be a prime ring with extended centroid C and two-sided Martindale quotient ring Q. Let d be a nonzero derivation of R, f(x ₁,…, x _n ) a multilinear polynomial over C, b ∈ R, and I a nonzero right ideal of R. Suppose that

for all r _i  ∈ I. If bI = 0, then either bd(I) = 0 or there exists an idempotent element e ∈ soc(RC) such that IC = eRC and f(x ₁,…, x _n ) is a PI for eRCe. If bI ≠ 0, then there exists an idempotent element e ∈ soc(RC) such that IC = eRC and one of the following holds:

(i) d = ad(a) for some a ∈ Q such that aI = 0 and f(x ₁,…, x _n ) ^k+1 is central- \p000valued on eRCe;

(ii) f(x ₁,…, x _n ) is central-valued in eRCe;

(iii) dim _C eRCe = 4 and char R = 2.

Key Words:

• Derivations
• Multilinear polynomials
• Prime rings
• Right ideals

2000 Mathematics Subject Classification:

• 16W25
• 16R20
• 16N60

ACKNOWLEDGMENTS

The author is very grateful to the referee for pointing out a gap in the original manuscript.

Notes

Communicated by M. Bresar.