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On topological lattices and their applications to module theory. (English) Zbl 1343.06005

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 \(\mathrm{Spec}^f(M)\).

MSC:

06B30 Topological lattices
16D10 General module theory in associative algebras
16D80 Other classes of modules and ideals in associative algebras
13C05 Structure, classification theorems for modules and ideals in commutative rings
13C13 Other special types of modules and ideals in commutative rings
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