The generalised RK-order, orthogonality and regular types for modules

Abstract

I characterise various model-theoretic properties of types, in complete theories of modules, in terms of the algebraic structure of pure-injective modules. More specifically, I consider the generalised RK-order, and the relation of domination between types, orthogonality of types, and regular types. It will be seen that, essentially, it is the stationary types over pure-injective models which bear the algebraic structural information.For background in the model theory of modules, see [4], [5], [13], [17], [19], and for the model-theoretic background [10], [11], [12], [18]. More specific references will be given in the text. I summarise some principal definitions and results below. First, though, let me describe the main results.Throughout this paper, R will be a ring with 1; the language will be that for (right R-) modules. All types will be complete types in a complete theory, T, of R-modules. The notation is reserved for an extremely saturated model of T, inside which all “small” situations may be found. Thus, sets of parameters are just subsets of , and all models will be elementary substructures of . A positive primitive, or pp, formula is one which is equivalent to a formula of the formwith the rij ∈ R.Suppose that p and q are 1-types over a pure-injective model M.