In this work we study the fragment structures over a ring extension $R$ of a ring $R_{0}$. The defining conditions of the fragments with the partial actions on the descending chains of $R_{0}$-modules measure how far they are from being \hbox{$R$-modules}. The category of $R$-fragments lies between the categories of $R_{0}$-modules and of \hbox{$R$-modules}. Inspite of $R$-fragments, in a general setting, are far from being \hbox{$R$-modules}; they behave, in some ways, the same as $R$-modules. We prove some imprtant results for finitely spanned fragments and some of their related properties.