We study various features of the lattice LMof 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 1- minimal topologies (=atoms in the subset of LMof 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 LMare 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.