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 ${\mathrm{\beth }}_{\alpha }(\mathrm{\mid }R{\mathrm{\mid }}^{<\theta })$ such that every parameter-free $L_{\mathrm{\infty },\theta }$ formula of rank at most $\alpha$ is equivalent in $M$ to an infinitary Boolean combination of cosets $\bar{a}+\phi (M)$, where $\bar{a}\in I_{\alpha }$ 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.