p-Convexly valued rings

In (J. Symbolic Logic 56(2) (1991) 539), Bélair developed a theory analogous to the theory of real closed rings in the p-adic context, namely the theory of p-adically closed integral rings. Firstly we use the property proved in Lemma 2.4 in (J. Symbolic Logic 60(2) (1995) 484) to express this theory in a language including a p-adic divisibility relation and we show that this theory admits definable Skolem functions in this language (in the sense of (J. Symbolic Logic 49 (1984) 625)). Secondly, we are interested in dealing with some questions similar to that of (Z. Math. Logik Grundlag. 29(5) (1983) 417); e.g., results about integral-definite polynomials over a p-adically closed integral ring A and a kind of “Nullstellensatz” using the notion of image-radical.