A product of two generalized derivations on polynomials in prime rings

• Autores: Vincenzo De Filippis
• Localización: Collectanea mathematica, ISSN 0010-0757, Vol. 61, Fasc. 3, 2010, págs. 302-322
• Idioma: español
• DOI: 10.1007/BF03191235
• Enlaces
  • Texto completo
• Resumen

  • 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(r_1, ..., r_n): r_1, ..., r_n ∃ 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:

    1. there existsα ∈ C such thatF(x)=αx, for allx ∈ R;

    2. there existsα ∈ C such thatG(x)=αx, for allx ∈ R;

    3. there exista; b ∈ U such thatF(x)=ax, G(x)=bx, for allx ∈ R;

    4. there exista; b ∈ U such thatF(x)=xa, G(x)=xb, for allx ∈ R;

    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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