Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

Artinian rings with nilpotent adjoint group

  • 34 Accesses

  • 1 Citations

Abstract

Let R be an Artinian ring (not necessarily with unit element), let Z(R) be its center, and let R° be the group of invertible elements of the ring R with respect to the operation ab = a + b + ab. We prove that the adjoint group R° is nilpotent and the set Z(R) + R° generates R as a ring if and only if R is the direct sum of finitely many ideals each of which is either a nilpotent ring or a local ring with nilpotent multiplicative group.

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

References

  1. 1.

    M. F. Atiyah and I. G. Macdonald, Introduction to Commutative Algebra, Addison-Wesley, Reading (1969).

  2. 2.

    I. Fisher and K. E. Eldridge, “D.C.C. rings with a cyclic group of units,” Duke Math. J., 34, 243–248 (1967).

  3. 3.

    I. Fisher and K. E. Eldridge, “Artinian rings with cyclic quasi-regular groups,” Duke Math. J., 36, No. 1, 43–47 (1969).

  4. 4.

    G. Groza, “Artinian rings having a nilpotent group of units,” J. Alg., 121, No. 2, 253–262 (1989).

  5. 5.

    N. Gupta and F. Levin, “On the Lie ideals of a ring,” J. Alg., 81, No. 1, 225–231 (1983).

  6. 6.

    N. Jacobson, Structure of Rings, American Mathematical Society (1964).

  7. 7.

    S. A. Jennings, “Radical rings with nilpotent associated groups,” Trans., Roy. Soc. Can., 49, No. 3, 31–38 (1955).

  8. 8.

    X. Du, “The centers of a radical ring,” Can. Math. Bull., 35, 174–179 (1992).

  9. 9.

    F. Catino and M. M. Miccoli, “Local rings whose multiplicative group is nilpotent,” Arch. Math. (Basel), 81, No. 2, 121–125 (2003).

  10. 10.

    B. Amberg and Ya. P. Sysak, “Associative rings whose adjoint semigroup is locally nilpotent,” Arch. Math. (Basel), 76, 426–435 (2001).

  11. 11.

    A. Mal’tsev, “Nilpotent semigroups,” Uch. Zap. Ivanov. Ped. Inst., Ser. Fiz.-Mat. Nauk., 4, 107–111 (1953).

  12. 12.

    I. Stewart, “Finite rings with a specified group of units,” Math. Z., 126, No. 1, 51–58 (1972).

  13. 13.

    M. I. Khuzurbazar, “Multiplicative group of a division ring,” Dokl. Akad. Nauk SSSR, 131, No. 6, 1268–1271 (1960).

Download references

Author information

Additional information

__________

Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 3, pp. 417–426, March, 2006.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Evstaf’ev, R.Y. Artinian rings with nilpotent adjoint group. Ukr Math J 58, 472–481 (2006). https://doi.org/10.1007/s11253-006-0079-4

Download citation

Keywords

  • Local Ring
  • Nilpotent Group
  • Unit Element
  • Division Ring
  • Multiplicative Group