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The classification of small weakly minimal sets. III: Modules

Published online by Cambridge University Press:  12 March 2014

Steven Buechler
Steven Buechler*
Affiliation:
Department of Mathematics, University of California, Berkeley, California 94720
*
Department of Mathematical Sciences, University of Wisconsin—Milwaukee, Milwaukee, Wisconsin 53201
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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 = acl(∅) ⋂ M and I = {r ∈ R: rM ⊂ A}. Notice that I is an ideal.

(i) F = R/Iis a finite field.

(ii) Suppose that a, b0,…,bn, ∈ M and . Then there are s, ri ∈ R, i ≤ n, such that sa + Σi≤nribi ∈ A and s ∉ 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).

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 53 , Issue 3 , September 1988 , pp. 975 - 979
Copyright
Copyright © Association for Symbolic Logic 1988

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References

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