  We study finitely generated models of countable theories, having at most
countably many nonisomorphic finitely generated models. We intro- duce 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 prop- erty 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.
