Sublattices of \(R\)-tors induced by \(\mathcal A\)-modules. (Q2920244)

For an associative ring with identity, \(R\), the authors consider the frame \(R\)-tors of all hereditary torsion theories in the category of left \(R\)-modules and, for a given \(\rho\in R\)-tors, they study the sublattice \(\mathrm{gen}(\rho)=\{\tau\in R\mathrm{-tors}\mid\tau\geq\rho\}\). This study is used to obtain new characterizations of \(\rho\)-Artinian rings, \(\rho\)-semiartinian rings, and rings with \(\rho\)-atomic dimension (rings such that the lattice \(\mathrm{gen}(\sigma)\) is atomic for all \(\sigma\geq\rho\)).