On rings with the serial embedding property

This thesis uses methods from ring and module theory. A ring is a set equipped with two operations satisfying properties analogous to those of addition and multiplication of integers. Modules are algebraic objects on which rings act. Rings and modules are examples of fundamental abstract structures and are subjects of central importance in one of the branches of mathematics called (modern abstract) algebra. Much of modern research in algebra is dedicated to studying rings. In a sense, the structure of a ring is reflected by the structure of its modules. In this thesis, we study rings under certain assumptions about their modules. More specifically, we consider the embedding of mod-ules into a specific class of modules constructed by, with certain constraints, combining modules whose submodules form a unique chain under inclusion, and we describe various rings and modules satisfying this embedding property. Moreover, we provide a new characterization of a class of rings called Dedekind domains.