We consider the reduct to the module language of certain theories of fields with a non surjective endomorphism. We show in some cases the existence of a model companion. We apply our results for axiomatizing the reduct to the theory of modules of non principal ultraproducts of separably closed fields of fixed but non zero imperfection degree.
"Asymptotic theory of modules of separably closed fields." J. Symbolic Logic 70 (2) 573 - 592, June 2005. https://doi.org/10.2178/jsl/1120224729