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Some identities related to multiplicative (generalized)-derivations in prime and semiprime rings

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Abstract

Let R be a semiprime ring with center Z(R) and \(\lambda \) a nonzero left ideal of R. A mapping \(F: R\rightarrow R\) (not necessarily additive) is said to be a multiplicative (generalized)-derivation on R, if there exists a map d (not necessarily an additive map or derivation) on R such that \(F(xy)=F(x)y+xd(y)\) holds for all \(x,y\in R\). Suppose that F and G are two multiplicative (generalized)-derivations of R associated with the maps d and g respectively on R. Throughout this paper we study the following situations:

  1. (1)

    \(F([x,y])+G(yx)+d(x)F(y)+xy \in Z(R)\),

  2. (2)

    \(F(x\circ y)+G(yx)+d(x)F(y)+xy \in Z(R)\),

  3. (3)

    \(F(xy)+G(yx)+d(x)F(y)\pm [x, y] \in Z(R)\),

  4. (4)

    \(F([x,y])+G(xy)+d(x)F(y)+yx \in Z(R)\),

  5. (5)

    \(F(x\circ y)+G(xy)+d(x)F(y)+yx \in Z(R)\),

  6. (6)

    \(F([x,y])+G(yx)+d(y)F(x)-xy \in Z(R)\),

  7. (7)

    \(F(x)F(y)-G(yx)-xy+yx \in Z(R)\); for all \(x,y\in \lambda \).

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References

  1. Albas, E.: Generalized derivations on ideals of prime rings. Miskolc Math. Notes 14(1), 3–9 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  2. Ali, S., Dhara, B., Dar, N.A., Khan, A.N.: On Lie ideals with multiplicative (generalized)-derivations in prime and semiprime rings. Beitr. Algebra Geom. 56(1), 325–337 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  3. Asma Ali, B., Dhara, S. Khan., Ali, F.: Multiplicative (Generalized)-derivations and left ideals in semiprime rings. Hacettep J. Math. Stat. 44(6), 1293–1306 (2015)

    MathSciNet  MATH  Google Scholar 

  4. Ali, A., De Filippis, V., Shujat, F.: On one sided ideals of a semiprime ring with generalized derivations. Aequat. Math. 85(3), 529–537 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  5. Ashraf, M., Ali, A., Ali, S.: Some commutativity theorems for rings with generalized derivations. Southeast Asian Bull. Math. 31, 415–421 (2007)

    MathSciNet  MATH  Google Scholar 

  6. Ashraf, M., Ali, A., Rani, R.: On generalized derivations of prime rings. Southeast Asian Bull. Math. 29, 669–675 (2005)

    MathSciNet  MATH  Google Scholar 

  7. Ashraf, M., Rehman, N.: On derivations and commutativity in prime rings. East-West J. Math. 3(1), 87–91 (2001)

    MathSciNet  MATH  Google Scholar 

  8. Daif, M.N., Tammam El-Sayiad, M.S.: Multiplicative generalized derivations which are additive. East-West J. Math. 9(1), 31–37 (1997)

    MathSciNet  MATH  Google Scholar 

  9. Daif, M.N., Bell, H.E.: Remarks on derivations on semiprime rings. Int. J. Math. Math. Sci. 15(1), 205–206 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  10. Dhara, B., Kar, S., Kuila, S.: A note on multiplicative (generalized)-derivations and left ideals in semiprime rings. Rend. Circ. Mat. Palermo II. Ser. 70(2), 631–640 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  11. Dhara, B., De Filippis, V.: Engel conditions of generalized derivations on left ideals and Lie ideals in prime rings. Comm. Algebra 48(1), 154–167 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  12. Dhara, B., Mozumder, M.R.: Some identities involving multiplicative generalized derivations in prime and semiprime rings. Bol. Soc. Paran. Mat. 36(1), 25–36 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  13. Dhara, B., Pradhan, K.G.: A note on multiplicative (generalized)-derivations with annihilator conditions. Georgian Math. J. 23(2), 191–198 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  14. Dhara, B., Pradhan, K.G.: A note on multiplicative (generalized)-derivations with annihilator conditions. Georgian Math. J. 23(2), 191–198 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  15. Dhara, B., Kar, S., Pradhan, K.G.: Generalized derivations acting as homomorphism or anti-homomorphism with central values in semiprime rings. Miskolc Math. Notes 16(2), 781–791 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  16. Dhara, B., Kar, S., Mondal, S.: A result on generalized derivations on Lie ideals in prime rings. Beitr. Algebra Geom. 54(20), 677–682 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  17. Dhara, B.: Generalized derivations acting as a homomorphism or anti-homomorphism in semiprime rings. Beitr. Algebra Geom. 53, 203–209 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  18. Dhara, B., Ali, S.: On multiplicative(generalized)-derivations in prime and semiprime rings. Aequat. Math. 86(1–2), 65–79 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  19. Dhara, B.: Remarks on generalized derivations in prime and semiprime rings. Int. J. Math. Math. Sci. 2010, 646587 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  20. Lee, T.K., Shiue, K.W.: A result on derivations with Engel condition in prime rings. Southeast Asian Bull. Math. 23, 437–446 (1999)

    MathSciNet  MATH  Google Scholar 

  21. Daif, M.N., Bell, H.E.: Remarks on derivations on semiprime rings. Internat. J. Math. Math. Sci. 15(1), 205–206 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  22. Tiwari, S.K., Sharma, R.K., Dhara, B.: Multiplicative (generalized)-derivations in semiprime rings. Beitr. Algebra Geom. 58(1), 211–225 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  23. Tiwari, S.K., Sharma, R.K., Dhara, B.: Identities related to generalized derivation on ideal in prime rings. Beitr. Algebra Geom. 57(4), 809–821 (2016)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

The authors would like to thank the referee for providing very helpful comments and suggestions. The third author is grateful to University Grants Commission, New Delhi, for JRF awarded to him under Grant No. F. 1133/(CSIR-UGC NET JUNE 2018) dated 16.04.2019.

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  1. Department of Mathematics, Belda College, Belda, Paschim Medinipur, West Bengal, 721424, India

    Basudeb Dhara

  2. Department of Mathematics, Jadavpur University, Kolkata, West Bengal, 700032, India

    Sukhendu Kar & Nripendu Bera

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  1. Basudeb Dhara
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Dhara, B., Kar, S. & Bera, N. Some identities related to multiplicative (generalized)-derivations in prime and semiprime rings. Rend. Circ. Mat. Palermo, II. Ser 72, 1497–1516 (2023). https://doi.org/10.1007/s12215-022-00743-w

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  • DOI: https://doi.org/10.1007/s12215-022-00743-w

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