Infinitary positive existential normal forms for modules
Let $\theta$ be a regular cardinal, $R$ a ring, and $M$ a left $R$-module. We prove that for every ordinal $\alpha$ there is a set $I_\alpha \subseteq M^{<\theta}$ of size at most $\beth_\alpha(|R| + \theta)$ such that every parameter-free $L_{\infty,\theta}$ formula of rank at most $\alpha$ is equivalent in $M$ to an infinitary Boolean combination of cosets $\overline{a} + \phi(M)$, where $\overline{a} \in I_\alpha$ and $\phi$ is an infinitary positive existential formula of rank at most $\alpha$. The main ingredient in the proof is a combinatorial lemma which says that given $\kappa$ subgroups of an abelian group, there is a set of at most $2^\kappa$ points which tests whether any family in which each member is either empty or a coset of the corresponding subgroup covers the whole group. The proof proceeds by applying this lemma fiberwise to show that the relevant complete Boolean algebras of positive-existentially definable cosets are closed under projections.