## A non structure theorem for an infinitary theory which has the unsuperstability property.(English)Zbl 0578.03020

Let $$\kappa$$, $$\lambda$$ be infinite cardinals, $$\psi \in L_{\kappa^+,\omega}$$. We say that the sentence $$\psi$$ has the $$\lambda$$-unsuperstability property if there are $$\{\phi_ n(\bar x,\bar y):$$ $$n<\omega \}$$ quantifier free first order formulas in L, a model M of $$\psi$$, and there exist $$\{$$ $$\bar a_{\eta}:$$ $$\eta \in^{\omega \geq}\lambda \}\subseteq | M|$$ satisfying: for all $$\eta \in^{\omega}\lambda$$, and for every $$\nu \in^{\omega >}\lambda$$, $$\nu <\eta \Leftrightarrow M\vDash \phi_{\ell (\nu)}[\bar a_{\nu},\bar a_{\eta}].$$
Theorem. Let $$\psi \in L_{\kappa^+,\omega}$$, $$\lambda$$ a Ramsey cardinal. If $$\psi$$ has the $$\lambda$$-unsuperstability property then for every cardinal $$\chi$$, $$\chi >| L| \cdot \aleph_ 0\Rightarrow I(\chi,\psi)=2^{\chi}$$. A more general theorem is proved; the proof uses a new partition theorem for trees.
An application of the theorem to the theory of modules is the following Corollary. Assume there exists a Ramsey cardinal. Let R be an integral domain. If $$DT_ R(=the$$ class of torsion divisible R-modules) has a structure theorem (i.e. there are few cardinal invariants such that every module can be characterized by the invariants) then R must be Noetherian. E.g. If every module from $$DT_ R$$ is a direct sum of countable generated modules then R is Noetherian.

### MSC:

 03C75 Other infinitary logic 03C45 Classification theory, stability, and related concepts in model theory 03C60 Model-theoretic algebra

### Keywords:

theory of modules; Ramsey cardinal