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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
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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

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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
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