대한수학회보 (Bulletin of the Korean Mathematical Society)

  • 제42권4호
  • /
  • Pages.679-690
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  • 2005
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  • 1015-8634(pISSN)
  • /
  • 2234-3016(eISSN)
대한수학회 (Korean Mathematical Society)
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A ONE-SIDED VERSION OF POSNER'S SECOND THEOREM ON MULTILINEAR POLYNOMIALS

  • 발행 : 2005.11.01
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초록

Let K be a commutative ring with unity, R a prime K-algebra of characteristic different from 2, d a non-zero derivation of R, I a non-zero right ideal of R, f($x_1,{\cdots},\;x_n$) a multilinear polynomial in n non-commuting variables over K, a $\in$ R. Supppose that, for any $x_1,{\cdots},\;x_n\;\in\;I,\;a[d(f(x_1,{\cdots},\;x_n)),\;f(x_1,{\cdots},\;x_n)]$ = 0. If $[f(x_1,{\cdots},\;x_n),\;x_{n+1}]x_{n+2}$ is not an identity for I and $$S_4(I,\;I,\;I,\;I)\;I\;\neq\;0$$, then aI = ad(I) = 0.

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참고문헌

  1. K. I. Beidar, W. S. Martindale III, and V. Mikhalev, Rings with generalized identities, Pure and Applied Math., Dekker, New York, 1996
  2. C. L. Chuang, GPIs having coefficients in Utumi quotient rings, Proc. Amer. Math. Soc. 103 (1988), 723-728
  3. C. M. Chang and T. K. Lee, Additive subgroup generated by polynomial values on right ideals, Comm. Algebra 29 (2001), no. 7, 2977-2984 https://doi.org/10.1081/AGB-100105000
  4. V. De Filippis and O. M. Di Vincenzo, Posner's second theorem and an annihilator condition, Math. Pannon. 12 (2001), no. 1, 69-81
  5. V. K. Kharchenko, Differential identities of prime rings, Algebra Logic 17 (1978), 154-168
  6. T. K. Lee, Derivations with Engel conditions on polynomials, Algebra Colloq. 5 (1998), no. 1, 13-24
  7. T. K. Lee, Semiprime rings with differential identities, Bull. Inst. Math. Acad. Sinica 20 (1992), 27-38
  8. U. Leron, Nil and power-central polynomials in rings, Trans. Amer. Math. Soc. 202 (1975), 97-103 https://doi.org/10.2307/1997300
  9. W. S. Martindale III, Prime rings satisfying a generalized polynomial identity, J. Algebra 12 (1969), 576-584 https://doi.org/10.1016/0021-8693(69)90029-5
  10. L. Rowen, Polynomial identities in ring theory, Pure and Applied Math. vol. 84, Academic Press, New York, 1980
  11. T. L. Wong, Derivations with power-central values on multilinear polynomials, Algebra Colloq. 4 (1996), no. 3, 369-378