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Year 2023, Volume: 34 Issue: 34, 48 - 61, 10.07.2023
https://doi.org/10.24330/ieja.1281636

Abstract

References

  • N. Argaç, L. Carini and V. De Filippis, An Engel condition with generalized derivations on Lie ideals, Taiwanese J. Math., 12(2) (2008), 419-433.
  • K. I. Beidar, Rings with generalized identities III, Vestnik Moskov. Univ. Ser. I Mat. Mekh, 4 (1978), 66-73.
  • K. I. Beidar, W. S. Martindale and A. V. Mikhalev, Rings with Generalized Identities, CRC Press, 1995.
  • M. Bresar, Centralizing mappings and derivations in prime rings, J. Algebra, 156(2) (1993), 385-394. C. L. Chuang, GPIs having coefficients in Utumi quotient rings, Proc. Amer. Math. Soc., 103(3) (1988), 723-728. V. De Filippis and O. M. Di Vincenzo, Vanishing derivations and centralizers of generalized derivations on multilinear polynomials, Comm. Algebra, 40(6) (2012), 1918-1932.
  • B. Dhara, Annihilator condition on power values of derivations, Indian J. Pure Appl. Math., 42 (2011), 357-369. O. M. Di Vincenzo, On the n-th centralizer of a Lie ideal, Boll. Un. Mat. Ital. A (7), 3 (1989), 77-85. T. Erickson, W. S. Martindale and J. Osborn, Prime nonassociative algebras, Pacific J. Math., 60(1) (1975), 49-63.
  • C. Faith and Y. Utumi, On a new proof of Litoff's theorem, Acta Math. Acad. Sci. Hungar., 14 (1963), 369-371.
  • I. N. Herstein, Topics in Ring Theory, University of Chicago Press, Chicago, 1969.
  • N. Jacobson, Structure of Rings, American Mathematical Society Colloquium Publications, 1956. V. K. Kharchenko, Differential identities of prime rings, Algebra i Logika, 17(2) (1978), 155-168. C. Lanski, An Engel condition with derivation, Proc. Amer. Math. Soc., 118(3) (1993), 731-734. C. Lanski and S. Montgomery, Lie structure of prime rings of characteristic 2, Pacific J. Math., 42 (1972), 117-136.
  • T. K. Lee, Semiprime rings with differential identities, Bull. Inst. Math. Acad. Sinica, 20(1) (1992), 27-38.
  • T. K. Lee, Generalized derivations of left faithful rings, Comm. Algebra, 27(8) (1999), 4057-4073.
  • W. S. Martindale, Prime rings satisfying a generalized polynomial identity, J. Algebra, 12 (1969), 576-584.
  • E. C. Posner, Derivations in prime rings, Proc. Amer. Math. Soc., 8 (1957), 1093-1100.
  • E. C. Posner, Prime rings satisfying a polynomial identity, Proc. Amer. Math. Soc., 11 (1960), 180-183. S. K. Tiwari, Identities with generalized derivations in prime rings, Rend. Circ. Mat. Palermo (2), 71 (2022), 207-223.

Two generalized derivations on Lie ideals in prime rings

Year 2023, Volume: 34 Issue: 34, 48 - 61, 10.07.2023
https://doi.org/10.24330/ieja.1281636

Abstract

Let $R$ be a prime ring of characteristic not equal to $2$, $U$ be the Utumi quotient ring of $R$ and $C$ be the extended centroid of $R$. Let $G$ and $F$ be two generalized derivations on $R$ and $L$ be a non-central Lie ideal of $R$. If $F\Big(G(u)\Big)u = G(u^{2})$ for all $u \in L$, then one of the following holds:

(1) $G=0$.
(2) There exist $p,q \in U$ such that $G(x)=p x$, $F(x)=qx$ for all $x \in R$ with $qp=p$.
(3) $R$ satisfies $s_4$.

References

  • N. Argaç, L. Carini and V. De Filippis, An Engel condition with generalized derivations on Lie ideals, Taiwanese J. Math., 12(2) (2008), 419-433.
  • K. I. Beidar, Rings with generalized identities III, Vestnik Moskov. Univ. Ser. I Mat. Mekh, 4 (1978), 66-73.
  • K. I. Beidar, W. S. Martindale and A. V. Mikhalev, Rings with Generalized Identities, CRC Press, 1995.
  • M. Bresar, Centralizing mappings and derivations in prime rings, J. Algebra, 156(2) (1993), 385-394. C. L. Chuang, GPIs having coefficients in Utumi quotient rings, Proc. Amer. Math. Soc., 103(3) (1988), 723-728. V. De Filippis and O. M. Di Vincenzo, Vanishing derivations and centralizers of generalized derivations on multilinear polynomials, Comm. Algebra, 40(6) (2012), 1918-1932.
  • B. Dhara, Annihilator condition on power values of derivations, Indian J. Pure Appl. Math., 42 (2011), 357-369. O. M. Di Vincenzo, On the n-th centralizer of a Lie ideal, Boll. Un. Mat. Ital. A (7), 3 (1989), 77-85. T. Erickson, W. S. Martindale and J. Osborn, Prime nonassociative algebras, Pacific J. Math., 60(1) (1975), 49-63.
  • C. Faith and Y. Utumi, On a new proof of Litoff's theorem, Acta Math. Acad. Sci. Hungar., 14 (1963), 369-371.
  • I. N. Herstein, Topics in Ring Theory, University of Chicago Press, Chicago, 1969.
  • N. Jacobson, Structure of Rings, American Mathematical Society Colloquium Publications, 1956. V. K. Kharchenko, Differential identities of prime rings, Algebra i Logika, 17(2) (1978), 155-168. C. Lanski, An Engel condition with derivation, Proc. Amer. Math. Soc., 118(3) (1993), 731-734. C. Lanski and S. Montgomery, Lie structure of prime rings of characteristic 2, Pacific J. Math., 42 (1972), 117-136.
  • T. K. Lee, Semiprime rings with differential identities, Bull. Inst. Math. Acad. Sinica, 20(1) (1992), 27-38.
  • T. K. Lee, Generalized derivations of left faithful rings, Comm. Algebra, 27(8) (1999), 4057-4073.
  • W. S. Martindale, Prime rings satisfying a generalized polynomial identity, J. Algebra, 12 (1969), 576-584.
  • E. C. Posner, Derivations in prime rings, Proc. Amer. Math. Soc., 8 (1957), 1093-1100.
  • E. C. Posner, Prime rings satisfying a polynomial identity, Proc. Amer. Math. Soc., 11 (1960), 180-183. S. K. Tiwari, Identities with generalized derivations in prime rings, Rend. Circ. Mat. Palermo (2), 71 (2022), 207-223.
There are 13 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Articles
Authors

Ashutosh Pandey This is me

Balchand Prajapati This is me

Early Pub Date May 11, 2023
Publication Date July 10, 2023
Published in Issue Year 2023 Volume: 34 Issue: 34

Cite

APA Pandey, A., & Prajapati, B. (2023). Two generalized derivations on Lie ideals in prime rings. International Electronic Journal of Algebra, 34(34), 48-61. https://doi.org/10.24330/ieja.1281636
AMA Pandey A, Prajapati B. Two generalized derivations on Lie ideals in prime rings. IEJA. July 2023;34(34):48-61. doi:10.24330/ieja.1281636
Chicago Pandey, Ashutosh, and Balchand Prajapati. “Two Generalized Derivations on Lie Ideals in Prime Rings”. International Electronic Journal of Algebra 34, no. 34 (July 2023): 48-61. https://doi.org/10.24330/ieja.1281636.
EndNote Pandey A, Prajapati B (July 1, 2023) Two generalized derivations on Lie ideals in prime rings. International Electronic Journal of Algebra 34 34 48–61.
IEEE A. Pandey and B. Prajapati, “Two generalized derivations on Lie ideals in prime rings”, IEJA, vol. 34, no. 34, pp. 48–61, 2023, doi: 10.24330/ieja.1281636.
ISNAD Pandey, Ashutosh - Prajapati, Balchand. “Two Generalized Derivations on Lie Ideals in Prime Rings”. International Electronic Journal of Algebra 34/34 (July 2023), 48-61. https://doi.org/10.24330/ieja.1281636.
JAMA Pandey A, Prajapati B. Two generalized derivations on Lie ideals in prime rings. IEJA. 2023;34:48–61.
MLA Pandey, Ashutosh and Balchand Prajapati. “Two Generalized Derivations on Lie Ideals in Prime Rings”. International Electronic Journal of Algebra, vol. 34, no. 34, 2023, pp. 48-61, doi:10.24330/ieja.1281636.
Vancouver Pandey A, Prajapati B. Two generalized derivations on Lie ideals in prime rings. IEJA. 2023;34(34):48-61.