Abstract
LetR be a prime ring of characteristic different from 2,U the Utumi quotient ring ofR, C the extended centroid ofR, F andG non-zero generalized derivations ofR andf(x 1, ...,x n ) a polynomial overC. Denote byf(R) the set {f(r1, ..., rn): r1, ..., rn ∃ R} of all the evaluations off(x 1, ...,x n ) inR. Suppose thatf(x 1, ...,x n ) is not central valued onR. IfR does not embed inM 2(K), the algebra of 2 × 2 matrices over a fieldK, and the composition (FG) acts as a generalized derivation on the elements off(R), then (FG) is a generalized derivation of R and one of the following holds:
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1.
there existsα ∈ C such thatF(x)=αx, for allx ∈ R;
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2.
there existsα ∈ C such thatG(x)=αx, for allx ∈ R;
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3.
there exista; b ∈ U such thatF(x)=ax, G(x)=bx, for allx ∈ R;
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4.
there exista; b ∈ U such thatF(x)=xa, G(x)=xb, for allx ∈ R;
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5.
there exista; b ∈ U, α,β ∈ C such thatF(x)=ax+xb, G(x)=αx+β(αx − xb), for allx ∈ R.
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De Filippis, V. A product of two generalized derivations on polynomials in prime rings. Collect. Math. 61, 303–322 (2010). https://doi.org/10.1007/BF03191235
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DOI: https://doi.org/10.1007/BF03191235