An algebraic dependence over the quasi-centre

  • 18 Accesses

  • 1 Citations


The authors study a class of rings generalizing both ξ-rings and rings which are (1) quasi-radical over their normalizer. The results are quite complete in the Noetherian case. In application several conditions on rings are shown to force the right duo condition. In particular it is shown that ξ-rings, algebras over the rationals satisfying (1), and algebraic algebras having no non central nilpotent elements (or a more general condition) are right duo or duo rings.


  1. [1]

    W. Burgess -M. Chacron,Subdirectly irreducible DQC rings, Canad. Math. Bull.,14 (4) (1941), pp. 495–497.

  2. [2]

    M. Chacron,On algebraic rings, Bull. Austr. Math. Soc.,1 (1969), pp. 385–389.

  3. [3]

    N. Divinsky,Commutative subdirectly irreducible rings, Proc. A.M.S.,8 (1957), pp. 642–648.

  4. [4]

    E. Feller,Properties of primary noncommutative rings, Trans. A.M.S.,89 (1958), pp. 79–91.

  5. [5]

    I. N. Herstein,The structure of a certain class of rings, Amer. J. Math.,75 (1953), pp. 866–871.

  6. [6]

    I. N. Herstein -R. Belluce -S. K. Jain,Generalized commutative rings, Nagoya Math. J.,27 (1966), pp. 1–5.

  7. [7]

    I. N. Herstein,Non commutative rings, The Carus Math. Monographs, no. 15, Open Court Publishing, Lasalle, Ill., 1968.

  8. [8]

    N. Jacobson,Structure of Rings, American Mathematical Society Colloquium Publications, vol.23 (1968), Providence, R.I.

  9. [9]

    L. Lesieur,Anneaux noethériens à gaughe complètement primaires et anneaux co-irréductibles, Reine Angew. (1969), pp. 106–117.

  10. [10]

    H. G. Moore,Subdirectly irreducible rings-some pathology, Bull. Austr. Math. Soc.,1 (1969), pp. 353–356.

  11. [11]

    L. W. Small,Orders in Artinian rings, J. of. Alg.,4 (1966), pp. 13–41.

  12. [12]

    G. Thierrin,Extensions radicales et quasi-radicales dans les anneaux, Canad. Math. Bull.,5 (1962), pp. 29–35.

  13. [13]

    G. Thierrin,On duo rings, Canad. Math. Bull.,3 (1960), pp. 167–172.

  14. [14]

    Y. Utumi,On ξ-rings, Osaka Math. J.,33 (1957), pp. 63–65.

  15. [15]

    A. P. J. van der Walt,Rings with dense quasi-centre, Math. Zeit.,97 (1967), pp. 38–44.

Download references

Author information

Additional information

Entrata in Redazione il 28 luglio 1972.

This research has been supported by grant A7876 of the National Research Council of Canada.

This research has been supported by grant A7877 of the National Research Council of Canada.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Chacron, M., Thierrin, G. An algebraic dependence over the quasi-centre. Annali di Matematica 110, 1–14 (1976).

Download citation


  • General Condition
  • Nilpotent Element
  • Algebraic Dependence
  • Algebraic Algebra
  • Noetherian Case