P-convexly valued rings

This work is a part of my research in the area of model theory. First we study a particular L-theory of rings whose models are called p-convexly valued domains (the first-order language L is the language of rings equipped with a linear divisibility predicate and predicates for the nth powers). It consists of a p-adic counterpart of Becker’s convexly oredered valuation rings. Then we are interested in a particular subclass of p-convexly valued domains, the p-adically closed integral rings (a p-adic counterpart of real closed valuation rings studied by G. Cherlin and M. Dickmann). In a first step we axiomatize the first-order L-theory whose models are elements of this class and prove some model-theoretic results i.e. the modelcompleteness of this L-theory, the existence of Skolem functions, a Cell decomposition theorem which used the work of topological L-structures studied by L. Mathews in his Ph.D. thesis. These previous results allow us to show some algebraic results for padically closed integral rings: