The classification of small weakly minimal sets. III: Modules

Journal of Symbolic Logic 53 (3):975-979 (1988)
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Abstract

Theorem A. Let M be a left R-module such that Th(M) is small and weakly minimal, but does not have Morley rank 1. Let $A = \mathrm{acl}(\varnothing) \cap M$ and $I = \{r \in R: rM \subset A\}$ . Notice that I is an ideal. (i) F = R/I is a finite field. (ii) Suppose that a, b 0 ,...,b n ∈ M and a b̄. Then there are s, r i ∈ R, i ≤ n, such that sa + ∑ i ≤ n r ib i ∈ A and $s \not\in I$ . It follows from Theorem A that algebraic closure in M is modular. Using this and results in [B1] and [B2], we obtain Theorem B. Let M be as in Theorem A. Then Vaught's conjecture holds for Th(M)

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References found in this work

Model theory of modules.Martin Ziegler - 1984 - Annals of Pure and Applied Logic 26 (2):149-213.
Superstable groups.Ch Berline & D. Lascar - 1986 - Annals of Pure and Applied Logic 30 (1):1-43.

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