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

Published online by Cambridge University Press:  12 March 2014

Mike Prest
Mike Prest*
Affiliation:
Mathematics Department, Yale University, New Haven, Connecticut 06520
*
Department of Pure Mathematics, University of Liverpool, Liverpool L69 3BX, England
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Extract

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 form

with the rij ∈ R.

Suppose that p and q are 1-types over a pure-injective model M.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 50 , Issue 1 , March 1985 , pp. 202 - 219
Copyright
Copyright © Association for Symbolic Logic 1985

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References

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