Some model theory over a nearly simple uniserial domain and decompositions of serial modules

By a careful investigation of the model theory of modules over a special class of uniserial domains we give some (counter) examples to a decomposition of a serial module. For instance, there is a uniserial module M over a uniserial domain that is not quasi-small. Also, there is a projective non-free countably generated module over the endomorphism ring of M.