Hostname: page-component-84b7d79bbc-fnpn6 Total loading time: 0 Render date: 2024-07-27T23:25:52.085Z Has data issue: false hasContentIssue false
  • English
  • Français

Chain Conditions for Modular Lattices with Finite Group Actions

Published online by Cambridge University Press:  20 November 2018

Joe W. Fisher
Joe W. Fisher*
Affiliation:
University of Cincinnati, Cincinnati, Ohio
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

This paper establishes the following combinatorial result concerning the automorphisms of a modular lattice.

THEOREM. Let M be a modular lattice and let G be a finite subgroup of the automorphism group of M. If the sublattice, MG, of (common) fixed points (under G) satisfies any of a large class of chain conditions, then M satisfies the same chain condition. Some chain conditions in this class are the following: the ascending chain condition; the descending chain condition; Krull dimension; the property of having no uncountable chains, no chains order-isomorphic to the rational numbers; etc.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 31 , Issue 3 , 01 June 1979 , pp. 558 - 564
Copyright
Copyright © Canadian Mathematical Society 1979

References

1. Isbell, J. R., Private communication with P. M. Neumann, 1969.Google Scholar
2. Gordon, R. and Robson, J. C., Krull dimension, Memoirs Amer. Math. Soc, 183 (1973).Google Scholar
3. Bergman, G. M., On chain conditions in modular lattices with finite group actions, after Isbell (unpublished note).Google Scholar
4. Bergman, G. M., On chain conditions in modular lattices with finite group actions, after Joe Fisher (unpublished note).Google Scholar
5. Fisher, J. W., Finiteness conditions for rings with finite group actions, Notices Amer. Math. Soc, 24 (1977), A–377.Google Scholar
6. Fisher, J. W. and Montgomery, S., Semiprime skew group rings, J. Algebra. 52 (1978), 241–247.Google Scholar
7. Fisher, J. W. and Osterburg, J., Semiprime ideals in rings with finite group actions, J. Algebra, to appear.Google Scholar
8. Fisher, J. W. and Osterburg, J., Some results on rings with finite group actions, Ring Theory: Proceedings of the Ohio Univ. Conf. (Marcel Dekker, 1976).Google Scholar
9. Reiter, E. E., Doctorial dissertation (U. of Cincinnati, 1978).Google Scholar
10. Formanek, E. and Jategaonker, A., Subrings of Noetherian rings, Proc. Amer. Math. Soc, 46 (1974), 181–186.Google Scholar