Skip to main content
SpringerLink
Log in

Derivations of group rings

  • Published:
Acta Scientiarum Mathematicarum Aims and scope Submit manuscript

Abstract

Let R[G] be the group ring of a group G over an associative ring R with unity such that all prime divisors of orders of elements of Gare invertible in R. If R is finite and G is a Chernikov (torsion FC-) group, then each R-derivation of R[G] is inner. Similar results also are obtained for other classes of groups G and rings R.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. H. ABU-KHUZAM, On rings with nil commutator ideal, Bull. Austral. Math. Soc. 23 (1981) 307–311, doi: 10.1017/S0004972700007152.

    MathSciNet  MATH  Google Scholar 

  2. H. AMBERG and Y. SYSAK, Associative rings with metabelian adjoint group, J. Algebra 277 (2004) 456–473, doi: 10.1016/j.jalgebra.2004.02.028.

    MathSciNet  MATH  Google Scholar 

  3. O. D. ARTEMOVYCH, Differentially trivial and rigid rings of finite rank, Period. Math. Hungar. 36 (1998) 1–16, doi: 10.1023/A:1004648818462.

    MathSciNet  MATH  Google Scholar 

  4. O. D. ARTEMOVYCH, Derivation rings of Lie rings, São Paulo J. Math. Sci. 13 (2019) 597–614, doi: 10.1007/s40863–017–0077–5.

    MathSciNet  MATH  Google Scholar 

  5. A. AL KHALAF, O. D. ARTEMOVYCH and I. TAHA, Derivations in differentially prime rings, J. Algebra Appl. 17 (2018), 1850129, 12, doi: 10.1142/S0219498818501293.

    MathSciNet  MATH  Google Scholar 

  6. A. AL KHALAF, O. D. ARTEMOVYCH and I. TAHA, Rings with simple Lie rings of Lie and Jordan derivations, J. Algebra Appl. 17 (2018), 1850078, 11, doi: 10.1142/S0219498818500780.

    MathSciNet  MATH  Google Scholar 

  7. Ch. BAI, D. MENG and S. HE, Derivations on Novikov algebras, Internat. J. Theoret. Phys. 42 (2003) 507–521, doi: 10.1023/A:1024437815600.

    MathSciNet  MATH  Google Scholar 

  8. K. BACLAWSKI, Automorphisms and derivations of incidence algebras, Proc. Amer. Math. Soc. 36 (1972) 351–356, doi: 10.2307/2039158.

    MathSciNet  MATH  Google Scholar 

  9. V. V. BAVULA, Derivations and skew derivations of the Grassmann algebras, J. Algebra Appl. 8 (2009) 805–827, doi: 10.1142/S0219498809003655.

    MathSciNet  MATH  Google Scholar 

  10. V. V. BAVULA, The group of automorphisms of the Lie algebra of derivations of a polynomial algebra, J. Algebra Appl. 16 (2017), 1750088, 8, doi: 10.1142/S0219498817500888.

    MATH  Google Scholar 

  11. H. E. BELL and A. A. KLEIN, Combinatorial commutativity and finiteness conditions for rings, Comm. Algebra 29 (2001) 2935–2943, doi: 10.1081/AGB-100104996.

    MathSciNet  MATH  Google Scholar 

  12. J. BERGEN, I. N. HERSTEIN and J. W. KERR, Lie ideals and derivations of prime rings, J. Algebra 71 (1981) 259–267, doi: 10.1016/0021–8693(81)90120–4.

    MathSciNet  MATH  Google Scholar 

  13. R. E. BLOCK, Determination of the differentiably simple rings with a minimal ideal, Ann. of Math. (2) 90 (1969) 433–459, doi: 10.2307/1970745.

    MathSciNet  MATH  Google Scholar 

  14. A. A. BOVDI, Group Rings, Kiev. UMK VO, 1988 (Russian).

    MATH  Google Scholar 

  15. A. A. BOVDI and I. I. KHRIPTA, Group algebras with a polycyclic multiplicative group, Ukrain. Mat. Zh. 38 (1986) 373–375, 407.

    MathSciNet  MATH  Google Scholar 

  16. V. D. BURKOV, Derivations of group rings, Abelian groups and modules, Tomsk. Gos. Univ., Tomsk, 1981, pp. 46–55, 246–247.

    Google Scholar 

  17. V. D. BURKOV, Derivations of polynomial rings, Vestnik Moskov. Univ. Ser. I Mat. Mekh. (1981), 51–55, 87.

    Google Scholar 

  18. I. G. CONNELL, On the group ring, Canad. J. Math. 15 (1963) 650–685, doi: 10. 4153/CJM-1963–067–0.

    MATH  Google Scholar 

  19. Ch. W. CURTIS and I. REINER, Representation theory of finite groups and associative algebras, Wiley Classics Library, John Wiley & Sons, Inc., New York, 1988.

    MATH  Google Scholar 

  20. F. DEMEYER and E. INGRAHAM, Separable algebras over commutative rings, Lecture Notes in Mathematics, Vol. 181, Springer-Verlag, Berlin-New York, 1971.

  21. M. FERRERO, A. GIAMBRUNO and C. P. MILIES, A note on derivations of group rings, Canad. Math. Bull. 38 (1995) 434–437, doi: 10.4153/CMB-1995–063–8.

    MathSciNet  MATH  Google Scholar 

  22. L. FUCHS, Infinite abelian groups. Vol. II, Pure and Applied Mathematics. Vol. 36-II, Academic Press, New York-London, 1973.

  23. N. GUPTA and F. LEVIN, On the Lie ideals of a ring, J. Algebra 81 (1983) 225–231, doi: 10.1016/0021–8693(83)90217-X.

    MathSciNet  MATH  Google Scholar 

  24. B. HARTLEY, The residual nilpotence of wreath products, Proc. London Math. Soc. (3) 20 (1970) 365–392, doi: 10.1112/plms/s3–20.3.365.

    MathSciNet  MATH  Google Scholar 

  25. I. N. HERSTEIN, Noncommutative rings, The Carus Mathematical Monographs, No. 15, Published by The Mathematical Association of America; distributed by John Wiley & Sons, Inc., New York, 1968.

    Google Scholar 

  26. N. JACOBSON, Abstract derivation and Lie algebras, Trans. Amer. Math. Soc. 42 (1937) 206–224, doi: 10.2307/1989656.

    MathSciNet  MATH  Google Scholar 

  27. S. A. JENNINGS, On rings whose associated Lie rings are nilpotent, Bull. Amer. Math. Soc. 53 (1947) 593–597, doi: 10.1090/S0002–9904–1947–08844–3.

    MathSciNet  MATH  Google Scholar 

  28. T. IKEDA, Locally inner derivations of ideally finite Lie algebras, Hiroshima Math. J. 17 (1987) 495–503.

    MathSciNet  MATH  Google Scholar 

  29. T. IKEDA and N. KAWAMOTO, On derivation algebras of group algebras, Non-associative algebra and its applications (Oviedo, 1993), Math. Appl., vol. 303, Kluwer Acad. Publ., Dordrecht, 1994, pp. 188–192.

    MathSciNet  MATH  Google Scholar 

  30. I. KAPLANSKY, Problems in the theory of rings. Report of a conference on linear algebras, June, 1956, pp. 1–3, National Academy of Sciences-National Research Council, Washington, Publ. 502, 1957.

    Google Scholar 

  31. I. KAYGORODOV and Yu. POPOV, A characterization of nilpotent nonassociative algebras by invertible Leibniz-derivations, J. Algebra 456 (2016) 323–347, doi: 10. 1016/j.jalgebra.2016.02.016.

    MathSciNet  MATH  Google Scholar 

  32. I. B. KAYGORODOV and Yu. S. POPOV, Alternative algebras that admit derivations with invertible values and invertible derivations, Izv. Ross. Akad. Nauk Ser. Mat. 78 (2014) 75–90.

    MathSciNet  MATH  Google Scholar 

  33. N. KAWAMOTO, Generalizations of Witt algebras over a field of characteristic zero, Hiroshima Math. J. 16 (1986) 417–426.

    MathSciNet  MATH  Google Scholar 

  34. V. K. KHARCHENKO, Automorphisms and derivations of associative rings, Mathematics and its Applications (Soviet Series), vol. 69, Kluwer Academic Publishers Group, Dordrecht, 1991.

  35. V. K. KHARCHENKO, Nekommutativnaya teoriya Galua, Nauchnaya Kniga (NII MIOONGU), Novosibirsk, 1996. With a biography of E. Galois in French.

    Google Scholar 

  36. G. KISS and M. LACZKOVICH, Derivations and differential operators on rings and fields, Ann. Univ. Sci. Budapest. Sect. Comput. 48 (2018) 31–43.

    MathSciNet  MATH  Google Scholar 

  37. A. N. KRASILNIKOV, On the group of units of a ring whose associated Lie ring is metabelian, Uspekhi Mat. Nauk 47 (1992) 217–218, doi: 10.1070/RM1992v047n06ABEH000973.

    MathSciNet  Google Scholar 

  38. CH. LANSKI, Subgroups and conjugates in semi-prime rings, Math. Ann. 192 (1971) 313–327, doi: 10.1007/BF02075359.

    MathSciNet  MATH  Google Scholar 

  39. CH. LANSKI and S. MONTGOMERY, Lie structure of prime rings of characteristic 2, Pacific J. Math. 42 (1972) 117–136.

    MathSciNet  MATH  Google Scholar 

  40. J. C. MCCONNELL and J. C. ROBSON, Noncommutative Noetherian Rings, Pure and Applied Mathematics (New York), John Wiley & Sons, Ltd., Chichester, 1987.

    MATH  Google Scholar 

  41. A. NOWICKI, Differential rings in which any ideal is differential, ActaUniv. Carolin. Math. Phys. 26 (1985), no. 2, 43–49.

    MathSciNet  MATH  Google Scholar 

  42. I. B. S. PASSI, D. S. PASSMAN and S. K. SEHGAL, Lie solvable group rings, Canad. J. Math. 25 (1973) 748–757, doi: 10.4153/CJM-1973–076–4.

    MathSciNet  MATH  Google Scholar 

  43. D. S. PASSMAN, Nil ideals in group rings, Michigan Math. J. 9 (1962) 375–384.

    MathSciNet  MATH  Google Scholar 

  44. D. S. PASSMAN, Infinite group rings, Marcel Dekker, Inc., New York, 1971. Pure and App. Math., 6.

    MATH  Google Scholar 

  45. M. PENKAVA and P. VANHAECKE, Deformation quantization of polynomial Poisson algebras, J. Algebra 227 (2000) 365–393, doi: 10.1006/jabr.1999.8239.

    MathSciNet  MATH  Google Scholar 

  46. D. J. S. ROBINSON, A course in the theory of groups, Second, Graduate Texts in Mathematics, vol. 80, Springer-Verlag, New York, 1996.

  47. C. K. SHARMA and S. SHRIVASTAVA, Some expansion formulae for the I-function, Proc. Nat. Acad. Sci. India Sect. A 62 (1992) 235–239.

    MathSciNet  MATH  Google Scholar 

  48. M. K. SMITH, Derivations of group algebras of finitely-generated, torsion-free, nilpotent groups, Houston J. Math. 4 (1978) 277–288.

    MathSciNet  MATH  Google Scholar 

  49. E. SPIEGEL, Derivations of integral group rings, Comm. Algebra 22 (1994) 2955–2959, doi: 10.1080/00927879408825003.

    MathSciNet  MATH  Google Scholar 

  50. J. B. SRIVASTAVA and R. K. SHARMA, Associated Lie algebras of group algebras, Algebra and its applications (New Delhi, 1981), Lecture Notes in Pure and Appl. Math., vol. 91, Dekker, New York, 1984, pp. 187–197.

    MathSciNet  MATH  Google Scholar 

  51. J. SZIGETI and L. VAN WYK, On Lie nilpotent rings and Cohen’s theorem, Comm. Algebra 43 (2015) 4783–4796, doi: 10.1080/00927872.2014.952735.

    MathSciNet  MATH  Google Scholar 

  52. X. TANG, L. LIU and J. XU, Lie bialgebra structures on derivation Lie algebra over quantum tori, Front. Math. China 12 (2017) 949–965, doi: 10.1007/s11464–017–0630–7.

    MathSciNet  MATH  Google Scholar 

  53. S. Tôgô, Derivations of Lie algebras, J. Sci. Hiroshima Univ. Ser. A-I Math. 28 (1964) 133–158.

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Applied Mathematics, Cracow University of Technology, Cracow, Poland

    Orest D. Artemovych

  2. UAEU, UAE

    Victor A. Bovdi

  3. UAEU, UAE

    Mohamed A. Salim

Authors
  1. Orest D. Artemovych
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Victor A. Bovdi
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Mohamed A. Salim
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Orest D. Artemovych.

Additional information

The research was supported by the UAEU UPAR grant G00002160.

Dedicated to the memory of Professor V. I. Sushchansky

Communicated by L. Molnár

Acknowledgment.

We would like to express our deep gratitude to the referee for the thoughtful and constructive review of our manuscript.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Artemovych, O.D., Bovdi, V.A. & Salim, M.A. Derivations of group rings. ActaSci.Math. 86, 51–72 (2020). https://doi.org/10.14232/actasm-019-664-x

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.14232/actasm-019-664-x

AMS Subject Classification (2010)

Key words and phrases

Search

Navigation