Published online by Cambridge University Press: 20 November 2018
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.