On Finitely Generated Models of Theories with at Most Countably Many Nonisomorphic Finitely Generated Models

We study finitely generated models of countable theories, having at most countably many nonisomorphic finitely generated models. We introduce a notion of rank of finitely generated models and we prove, when T has at most countably many nonisomorphic finitely generated models, that every finitely generated model has an ordinal rank. This rank is used to give a property of finitely generated models analogue to the Hopf property of groups and also to give a necessary and sufficient condition for a finitely generated model to be prime of its complete theory. We investigate some properties of limit groups of equationally noetherian groups, in respect to their ranks.