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The Hausdorff Completion of the Space of Closed Subsets of a Module

Published online by Cambridge University Press:  20 November 2018

E. W. Johnson  and
Johnny A. Johnson
E. W. Johnson
Affiliation:
Department of Mathematics, University of Iowa, Iowa City, Iowa 52242, U.S.A
Johnny A. Johnson
Affiliation:
Department of Mathematics, University of Houston, Houston, Texas 77204-3476, U.S.A.
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Abstract

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In this paper, we show that the lattice of closed subsets of the completion, in the Jacobson radical topology, of a finitely generated module M is isomorphic to the completion, under the Hausdorff topology, of the lattice of closed subsets of M. This extends submodule-theoretic results for complete modules to modules satisfying Chevalley's Theorem. We show that the lattice of submodules of every finitely generated module over a semi-local ring R is complete in the Hausdorff topology if and only if R is complete in the Jacobson radical topology.

Keywords

13A1513C9953
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 38 , Issue 3 , 01 September 1995 , pp. 325 - 329
Copyright
Copyright © Canadian Mathematical Society 1995

References

1. Anderson, D. D., The existence of dual modules, Proc. Amer. Math. Soc. 55(1976), 258–260.Google Scholar
2. Lu, Chin-Pi, Quasi-Complete Modules, Indiana Univ. Math. J. 29(1980), 277–286.Google Scholar
3. Nagata, M., Local Rings, Wiley-Intersci. Publ, New York, 1962.Google Scholar
4. Northcott, D. G., Lessons on Rings, Modules and Multiplicities, Cambridge Univ. Press, 1968.Google Scholar
5. Zariski, Oscar and Samuel, Pierre, Commutative Algebra, vol. 2, Van Nostrand, New York, 1960.Google Scholar