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

摘要

In this paper, we show that the lattice of closed subsets of the comple-tion, in the Jacobson radical topology, of a finitely generated module M is isomorphic to the completion, under the Hausdorif 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 gener-ated 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.