Sublattices of the lattice of pre-natural classes of modules
Let \(R\) be a ring with identity. A class of right \(R\)-modules is a pre-natural class if it is closed under isomorphic copies, submodules, arbitrary direct sums, and certain essential extensions. The collection \({\mathcal N}^p_r(R)\) of all pre-natural classes was invented by \textit{Y. Zhou} [Commun. Algebra 25, No. 2, 543-557 (1997; Zbl 0871.16016)] and studied in detail [in J. Pure Appl. Algebra 140, No. 2, 191-207 (1999; Zbl 0947.16003)]. This is a lattice, which contains many previously studied lattices of \(R\)-module classes such as \(M\)-natural classes for a fixed module \(M\), natural classes, and hereditary pre-torsion classes. In this paper the sublattice structure of \({\mathcal N}^p_r(R)\) is studied and is related to ring and module properties of \(R\). Let \({\mathcal T}^p_r(R)\) be the set of all hereditary pre-torsion classes. For an arbitrary pre-natural class \(\mathcal K\), let \({\mathcal T}^p_r({\mathcal K},R)=\{{\mathcal K}\cap{\mathcal T}:{\mathcal T}\in{\mathcal T}^p_r(R)\}\). Then it is shown that some of the theory of \({\mathcal T}^p_r(R)\) is extended to \({\mathcal T}^p_r({\mathcal K},R)\). Indeed this is a complete sublattice of \({\mathcal N}^p_r(R)\) and is algebraic and upper continuous. For the set \({\mathcal N}_r(R)\) of all natural classes, let \({\mathcal N}(R,{\mathcal K})=\{{\mathcal L}\cap{\mathcal K}:{\mathcal L}\in{\mathcal N}_r(R)\}\). Then this is also sublattice of \({\mathcal N}^p_r(R)\) and is a complete Boolean lattice. Let \({\mathcal T}_r(R)\) be the set of all hereditary torsion classes. Handelman and Viola-Prioli have conjectured that \(R\) is right Noetherian if \({\mathcal T}^p_r(R)={\mathcal T}_r(R)\). Fenrick proved that if \(R\) is right Noetherian such that \({\mathcal T}^p_r(R)={\mathcal T}_r(R)\), then \(R\) is a right V-ring. In spite of Fenrick's result, even in the Noetherian case the characterization of the rings \(R\) for which \({\mathcal T}^p_r(R)={\mathcal T}_r(R)\) remains open. The authors show that \(R\) is right Noetherian with \({\mathcal T}^p_r(R)={\mathcal T}_r(R)\) if and only if \(R\) is a right QI-ring. Finally, the authors show that \({\mathcal N}^p_r(\cdot)\) is a contravariant functor. One advantage of a functional approach is that it identifies several complete sublattices that \({\mathcal N}^p_r(R)\) must necessarily have. Some of the known functoriality of \({\mathcal N}_r(\cdot)\) is extended to \({\mathcal N}(\;,{\mathcal K})\).
scientific article; zbMATH DE number 1525269