A representation theorem for certain Boolean lattices (Q1106250)

Let R be an associative ring with 1 and R-tors the complete Brouwerian lattice of all hereditary torsion theories on the category of left R- modules. A well known result asserts that R is a left semiartinian ring iff R-tors is a complete atomic Boolean lattice. In this note, we prove that if \({\mathcal L}\) is a complete atomic Boolean lattice then there exists a left semiartinian ring R such that \({\mathcal L}\) is lattice-isomorphic to R-tors.