Geometrically closed rings

We develop the basic theory of geometrically closed rings as a generalisation of algebraically closed fields, on the grounds of notions coming from positive model theory and affine algebraic geometry. For this purpose we consider several connections between finitely presented rings and ultraproducts, affine varieties and definable sets, and we introduce the key notion of an arithmetic theory as a purely algebraic version of coherent logic for rings.Comment: 18 page

Positive model theory and amalgamations

This article has been accepted at Notre Dame Journal of Formal Logic.We continue the analysis of foundations of positive model theory as introduced by Ben Yaacov and Poizat. The objects of this analysis are h-inductive theories and their models, especially the \positively" ex- istentially closed ones. We analyze topological properties of spaces of types, introduce forms of quantifier elimination and characterize mini- mal completions of arbitrary h-inductive theories. The main technical tools consist of various forms of amalgamations in special classes of structures