On lattices associated to rings with respect to a preradical

Abstract

We introduce some new lattices of classes of modules with respect to appropriate preradicals. We introduce some concepts associated with these lattices, such as the $\sigma$-semiartinian rings, the $\sigma$-retractable modules, the $\sigma$-$V$-rings, the $\sigma$-max rings. We continue to study $\sigma$-torsion theories, $\sigma$-open classes, $\sigma$-stable classes. We prove some theorems that extend some known results. Our results fall into well known situations when the preradical $\sigma$ is chosen as the identity preradical.

Keywords

• Preradical
• lattice of module class
• hereditary torsion class
• boolean lattice
• V-ring
• max-ring

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