GENERALIZED SKEW DERIVATIONS ON MULTILINEAR POLYNOMIALS IN RIGHT IDEALS OF PRIME RINGS

Abstract

Let R be a prime ring, f (x, . . . , xn ) a multilinear polynomial over Cin n noncommuting indeterminates, I a nonzero right ideal of R, andF : R → R be a nonzero generalized skew derivation of R.Suppose that F (f (r, . . . , rn ))f (r, . . . , rn ) ∈ C, for all r, . . . , rn ∈ I.If f (x, . . . , xn ) is not central valued on R, then either char(R) = 2and R satisfies sor one of the following holds:(i ) f (x, . . . , xn )x n+1 is an identity for I;(ii ) F (I)I = (0);(iii ) [f (x, . . . , xn ), x n+1 ]x n+2 is an identity for I, there existb, c, q ∈ Q with q an invertible element such that F (x) =bx − qxq−1 c for all x ∈ R, and q−1 cI ⊆ I.Moreover, inthis case either (b − c)I = (0) or b − c ∈ C and f (x, . . . , xn ) is central valued on R.

Keywords

• Identity, generalized skew derivation, automorphism, (semi-)prime ring.

GENERALIZED SKEW DERIVATIONS ON MULTILINEAR POLYNOMIALS IN RIGHT IDEALS OF PRIME RINGS

Abstract

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Keywords

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