Infinitary positive existential normal forms for modules

Abstract

Let $θ$ be a regular cardinal, $R$ a ring, and $M$ a left $R$-module. We prove that for every ordinal $α$ there is a set $I_α\subseteq M^{<θ}$ of size at most $\beth_α(|R| + θ)$ such that every parameter-free $L_{\infty,θ}$ formula of rank at most $α$ is equivalent in $M$ to an infinitary Boolean combination of cosets $\overline{a} + ϕ(M)$, where $\overline{a} \in I_α$ and $ϕ$ is an infinitary positive existential formula of rank at most $α$. The main ingredient in the proof is a combinatorial lemma which says that given $κ$ subgroups of an abelian group, there is a set of at most $2^κ$ 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.