## Abstract

A prime model <jats:italic>O</jats:italic> of some complete theory <jats:italic>T</jats:italic> is a model which can be elementarily imbedded into any model of <jats:italic>T</jats:italic> (cf. Vaught [7, Introduction]). We are going to replace the assumption that <jats:italic>T</jats:italic> is complete and that the maps between the models of <jats:italic>T</jats:italic> are elementary imbeddings (elementary extensions) by more general conditions. <jats:italic>T</jats:italic> will always be a first order theory with identity and may have function symbols. The language <jats:italic>L(T)</jats:italic> of <jats:italic>T</jats:italic> will be denumerable. The maps between models will be so called <jats:italic>F</jats:italic>-maps, i.e. maps which preserve a certain set <jats:italic>F</jats:italic> of formulas of <jats:italic>L(T)</jats:italic> (cf. I.1, 2). Roughly speaking a generalized prime model of <jats:italic>T</jats:italic> is a denumerable model <jats:italic>O</jats:italic> which permits an <jats:italic>F</jats:italic>-map <jats:italic>O→M</jats:italic> into any model <jats:italic>M</jats:italic> of <jats:italic>T</jats:italic>. Furthermore <jats:italic>O</jats:italic> has to be “generated” by formulas which belong to a certain subset <jats:italic>G</jats:italic> of <jats:italic>F</jats:italic>.