On rings whose modules embed into serial modules

We study rings which have the property that certain classes of modules embed into direct sum of uniserial modules (serial modules). We study when all modules embed into serial modules, respectively when finitely generated or when torsion modules embed into serial modules. We give a number of equivalent characterizations of these properties, and in particular, re-obtain results on rings whose modules embed into serial modules from Tuganbaev. We fully classify commutative rings all of whose torsion modules embed into serials, and show they are either Dedekind domains or finite products of discrete valuation domains and Artinian chain rings. In particular, this provides a new characterization of Dedekind domains.