ON THE LATTICE OF LINEAR MODULE TOPOLOGIES

We study various features of the lattice ${\cal L}_M$ of linear module topologies on a module $M$ and their impact on the structure of the abstract module. A particular emphasis is given to: a) permanence properties of $l$-minimal topologies (=atoms in the subset of ${\cal L}\!_M$ of Hausdorff topologies) in analogy with the theory of minimal topological groups; b) the usual equivalence between linear topologies and the equivalence classes of the atoms; c) characterization of the abstract modules $M$ such that certain classes in ${\cal L}\!_M$ are singletons (in particular, modules such that each non-discrete linear topology is topologically artinian). Point c) involves the class of modules having all proper quotients artinian.