Modules with an almost categorical theory (Q1070224)

A complete theory is called almost categorical [\textit{O. V. Belegradek}, Sib. Math. Zh. 14, 277-288 (1973; Zbl 0262.02047)] if it is totally transcendental and a certain principal extension of it has a finite set of strongly minimal formulas whose disjunction is non-two-cardinal. We consider left modules over R (an associative ring with unity) as algebraic systems of the signature \(<+,0,r>_{r\in R}\) where r are symbols of unary operations, and give a characterization of modules with almost categorical elementary theory. For a module M let us denote by \(T_ M\) its elementary theory, \(E_ M\) be the class of all indecomposable, algebraically compact modules U such that \(M\oplus U\equiv M\), and \(\hat E_ M\) be the set of isomorphism types for modules from \(E_ M.\) Theorem. The theory \(T_ M\) is almost categorical if and only if \(T_ M\) is totally transcendental and the following conditions are fulfilled: 1) \(| \tilde E_ M| <\omega\); 2) for every \(U\in E_ M\), there exists a positively primitive formula \(\phi_ U(x)\) such that for \(U'\in E_ M:\) \(\phi_ U(U')\neq 0\Leftrightarrow U'\simeq U.\) Corollary. Let G be an abelian group, \[ \check S(G)=\oplus_{p\in P}\oplus_{k\in \omega}Z(p^{(k)})^{(\alpha_{p,k})}\oplus \oplus_{p\in P}Z(p^{\infty})^{(\beta_ p)}\oplus \oplus_{p\in P}R_ p^{(\gamma_ p)}\oplus Q^{(\epsilon)} \] be its Shmelev canonical form. The group G is almost categorical if and only if the following conditions are fulfilled: 1) \(\gamma_ p=0\) for all \(p\in P\); 2) \(\beta_ p<\omega\) for all \(p\in P\); 3) \(\alpha_{p,k}=\omega\) only for a finite number of pairs (p,k).