We introduce and study a local global principle for rings called LGPH, analogous to the nonsingular Hasse principle. The class of rings satisfying LGPH has been proved to contain many rings which occur naturally, such as the integral closure of the ring of integers inside any local field. We prove that LGPH is recursively axiomatisable in the language of rings, and derive from that a natural recursive axiomatisation of the ring of all \(p\)-adic algebraic integers, that is the integral closure of the ring of integers inside the field of \(p\)-adic numbers. Moreover the theory of that ring in a definable enrichment of the language of rings (one binary relation symbol being added) is proved to be model-complete. As an application we obtain that this ring has a decidable complete theory.