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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

Information

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

References

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