Hostname: page-component-84b7d79bbc-tsvsl Total loading time: 0 Render date: 2024-07-28T05:19:07.104Z Has data issue: false hasContentIssue false
  • English
  • Français

Localization of modular lattices, Krull dimension, and the Hopkins-Levitzki theorem (I)

Published online by Cambridge University Press:  24 October 2008

Toma Albu  and
Patrick F. Smith
Toma Albu
Affiliation:
Facultatea de Matematicač. Universitatea Bucureşti Str. Academiei 14, RO-70109 Bucharest 1, Romania e-mail: talbu@imar.ro
Patrick F. Smith
Affiliation:
Department of Mathematics, University of Glasgow. Glasgow G12 8QW e-mail: pfs@maths.gla.ac.uk
Get access Rights & Permissions [Opens in a new window]

Extract

The Hopkins–Levitzki Theorem, discovered independently in 1939 by C. Hopkins and J. Levitzki states that a right Artinian ring with identity is right Noetherian. In the 1970s and 1980s it has been generalized to modules over non-unital rings by Shock[10], to modules satisfying the descending chain condition relative to a heriditary torsion theory by Miller-Teply[7], to Grothendieck categories by Năstăsescu [8], and to upper continuous modular lattices by Albu [1]. The importance of the relative Hopkins-Levitzki Theorem in investigating the structure of some relevant classes of modules, including injectives as well as projectives is revealed in [3] and [6], where the main body of both these monographs deals with this topic. A discussion on the various forms of the Hopkins–Levitzki Theorem for modules and Grothendieck categories and the connection between them may be found in [3].

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 120 , Issue 1 , July 1996 , pp. 87 - 101
Copyright
Copyright © Cambridge Philosophical Society 1996

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Albu, T.. Certain Artinian lattices are Noetherian. Applications to the relative Hopkins-Levitzki theorem. In Methods in ring theory, edited by Van Oystaeyen, F. (D. Reidel Publishing, 1984), pp. 37–52.Google Scholar
[2]Albu, T.. On composition series of a module with respect to a set of Gabriel topologies. In Abelian Groups and Modules, edited by Göbel, R., Metelli, C., Orsatti, A., Salce, L. (Springer-Verlag, 1984), pp. 467–476.Google Scholar
[3]Albu, T. and Năstăsescu, C.. Relative finiteness in module theory (Marcel Dekker, Inc., 1984).Google Scholar
[4]Bican, L., Kepka, T. and Nemec, P.. ‘Rings, modules and preradicals’ (Marcel Dekker, Inc., 1982).Google Scholar
[5]Crawley, P. and Dilworth, R. P.. Algebraic theory of lattices (Prentice-Hall, Inc., 1973).Google Scholar
[6]Faith, C.. Injective modules and injective quotient rings (Marcel Dekker, Inc., 1982).Google Scholar
[7]Miller, R. W. and Teply, M. L.. The descending chain condition relative to a torsion theory. Pacific J. Math. 83 (1979), 207–220.Google Scholar
[8]Năstăsescu, C.. Conditions de finitude pour les modules. Rev. Roumaine Math. Pures Appl. 24 (1979), 745–758.Google Scholar
[9]Puczylowski, E. R.. Dimensions of modular lattices (Universidade Federal do Rio Grande do Sul, Brasil, 1987).Google Scholar
[10]Shock, R. C.. Certain Artinian rings are Noetherian. Canad. J. Math. 24 (1972), 553–556.CrossRefGoogle Scholar
[11]Stenström, B.. Rings of quotients (Springer-Verlag, 1975).CrossRefGoogle Scholar