On topological lattices and their applications to module theory. (Q2797001)

In this quite interesting paper, the authors study topological lattices relative to a proper subset of the lattice and study its properties. Next apply these results to study properties of modules. After introduction and preliminaries in Section 1, in Section 2, for a complete lattice \(\mathcal L=(L,\wedge,\vee,0,1)\) and a proper subset \(X\) of \(L-\{1\}\), they consider \(X\)-top lattices. Their main theorem in this section is that \(\mathcal L\) is an \(X\)-top lattice if and only if every element of \(X\) is strongly irreducible in \((C(L),\wedge)\). In Section 3, they introduce prime modules and first submodules of a module, as a dual to second submodules, and derive some equivalent conditions for a first submodule of a module. In the main Section 4, for an \(R\)-module \(M\), they define a topology on the set \(\mathrm{Spec}^f(M)\) of all first submodules of \(M\). If with this topology the dual lattice \(\mathcal L(M)^\circ\) is a \(\mathrm{Spec}^f(M)\)-top lattice then they call \(M\) to be a \(\mathrm{top}^f\)-module. If every first submodule of \(M\) is strongly hollow then \(M\) is said to be strongly \(\mathrm{top}^f\)-module. The authors derive some module theoretic properties of \(\mathrm{top}^f\)-modules and strongly \(\mathrm{top}^f\)-modules \(M\) in terms of topological properties of \(\mathrm{Spec}^f(M)\).