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Annihilators of power values of b-generalized derivations in prime rings

Year 2020, Volume: 69 Issue: 2, 1278 - 1284, 31.12.2020
https://doi.org/10.31801/cfsuasmas.628755

Abstract

Let $R$ be a prime ring with extended centroid $C$ and maximal left ring of quotients $Q_{ml}(R)$.  For  a nonzero element $b\in R$ let $F:R\rightarrow R$ be a right generalized $b$-derivation associated with the map $d$ of $R$. Suppose that  $s\left(F(x)\right)^n=0$ for all $x\in R$ where $s$ is a nonzero element in $R$ and $n\geq 1$ is a fixed positive integer. Then  there exist some $c\in Q{ml}(R)$ and $\beta \in C$ such that $d(x)=ad_c(x)$, $F(x)=(b+\beta)xb$ for all $x\in R$ and either $s(c+\beta)=0$ or $b(c+\beta)=0$.

References

  • Beidar, K. I., Martindale, W. S. III, Mikhalev, A. V., Rings with generalized identities, Marcel Dekker, Inc., New York, xiv+522 pp, 1996.
  • Brešar, M., A note on derivations, Math. J. Okayama Univ., 32 (1990), 83-88.
  • Chang, J.C., Annihilators of power values of a right generalized (α,β)-derivation, Bull. Inst. Math. Acad. Sin., 4 (1) (2009), 67-73.
  • Chuang, C. L., GPIs having coefficients in Utumi quotient rings, Proc. Amer. Math. Soc., 103 (3) (1988), 723-728.
  • Erickson, T. S.; Martindale, W. S., 3rd; Osborn, J. M., Prime nonassociative algebras, Pacific J. Math., 60 (1975), No. 1, 49-63.
  • Faith, C., Utumi, Y., On a new proof of Litoff's theorem, Acta Math. Acad. Sci. Hung., 14 (1967), 369-371.
  • Felzenszwalb, B., On a result of Levitzki, Canad. Math. Bull. 21 (2) (1978), 241-242.
  • Giambruno, A., Herstein, I.N., Derivations with nilpotent values, Rend. Circ. Mat. Palermo, 30 (2) (1981), 199-206.
  • Kharchenko,V.K., Differential identities of semiprime rings, Algebra Logika, 18 (1979),86-119; Algebra Logic, (English translation), 18 (1979), 58-80.
  • Kosan, M. T., Lee, T.K., b-generalized derivations having nilpotent values, J. Aust. Math. Soc., 96 (3) (2014), 326-337.
  • Lee, T.K., Lin, J.S., A result on derivations, Proc. Amer. Math. Soc., 124 (1996), 1687-1691.
  • Lee, T. K., Generalized derivations of left faithful rings, Comm. Algebra, 27(8) (1999), 4057-4073.
  • Martindale, W. S. 3rd, Prime rings satisfying a generalized polynomial identity, J. Algebra, 12 (1969), 576-584.
Year 2020, Volume: 69 Issue: 2, 1278 - 1284, 31.12.2020
https://doi.org/10.31801/cfsuasmas.628755

Abstract

References

  • Beidar, K. I., Martindale, W. S. III, Mikhalev, A. V., Rings with generalized identities, Marcel Dekker, Inc., New York, xiv+522 pp, 1996.
  • Brešar, M., A note on derivations, Math. J. Okayama Univ., 32 (1990), 83-88.
  • Chang, J.C., Annihilators of power values of a right generalized (α,β)-derivation, Bull. Inst. Math. Acad. Sin., 4 (1) (2009), 67-73.
  • Chuang, C. L., GPIs having coefficients in Utumi quotient rings, Proc. Amer. Math. Soc., 103 (3) (1988), 723-728.
  • Erickson, T. S.; Martindale, W. S., 3rd; Osborn, J. M., Prime nonassociative algebras, Pacific J. Math., 60 (1975), No. 1, 49-63.
  • Faith, C., Utumi, Y., On a new proof of Litoff's theorem, Acta Math. Acad. Sci. Hung., 14 (1967), 369-371.
  • Felzenszwalb, B., On a result of Levitzki, Canad. Math. Bull. 21 (2) (1978), 241-242.
  • Giambruno, A., Herstein, I.N., Derivations with nilpotent values, Rend. Circ. Mat. Palermo, 30 (2) (1981), 199-206.
  • Kharchenko,V.K., Differential identities of semiprime rings, Algebra Logika, 18 (1979),86-119; Algebra Logic, (English translation), 18 (1979), 58-80.
  • Kosan, M. T., Lee, T.K., b-generalized derivations having nilpotent values, J. Aust. Math. Soc., 96 (3) (2014), 326-337.
  • Lee, T.K., Lin, J.S., A result on derivations, Proc. Amer. Math. Soc., 124 (1996), 1687-1691.
  • Lee, T. K., Generalized derivations of left faithful rings, Comm. Algebra, 27(8) (1999), 4057-4073.
  • Martindale, W. S. 3rd, Prime rings satisfying a generalized polynomial identity, J. Algebra, 12 (1969), 576-584.
There are 13 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Research Articles
Authors

Nihan Baydar Yarbil 0000-0003-1376-2349

Publication Date December 31, 2020
Submission Date October 3, 2019
Acceptance Date July 20, 2020
Published in Issue Year 2020 Volume: 69 Issue: 2

Cite

APA Baydar Yarbil, N. (2020). Annihilators of power values of b-generalized derivations in prime rings. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, 69(2), 1278-1284. https://doi.org/10.31801/cfsuasmas.628755
AMA Baydar Yarbil N. Annihilators of power values of b-generalized derivations in prime rings. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. December 2020;69(2):1278-1284. doi:10.31801/cfsuasmas.628755
Chicago Baydar Yarbil, Nihan. “Annihilators of Power Values of B-Generalized Derivations in Prime Rings”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 69, no. 2 (December 2020): 1278-84. https://doi.org/10.31801/cfsuasmas.628755.
EndNote Baydar Yarbil N (December 1, 2020) Annihilators of power values of b-generalized derivations in prime rings. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 69 2 1278–1284.
IEEE N. Baydar Yarbil, “Annihilators of power values of b-generalized derivations in prime rings”, Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat., vol. 69, no. 2, pp. 1278–1284, 2020, doi: 10.31801/cfsuasmas.628755.
ISNAD Baydar Yarbil, Nihan. “Annihilators of Power Values of B-Generalized Derivations in Prime Rings”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 69/2 (December 2020), 1278-1284. https://doi.org/10.31801/cfsuasmas.628755.
JAMA Baydar Yarbil N. Annihilators of power values of b-generalized derivations in prime rings. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 2020;69:1278–1284.
MLA Baydar Yarbil, Nihan. “Annihilators of Power Values of B-Generalized Derivations in Prime Rings”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, vol. 69, no. 2, 2020, pp. 1278-84, doi:10.31801/cfsuasmas.628755.
Vancouver Baydar Yarbil N. Annihilators of power values of b-generalized derivations in prime rings. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 2020;69(2):1278-84.

Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics.

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