Recall that a (unital) Boolean ring is a (unital) commutative ring $A$ where every element is idempotent; it follows that $A$ is of characteristic 2. There is an equivalence of categories between Boolean rings and Boolean algebras; the Boolean algebra corresponding to a Boolean ring $A$ (which I'll continue to call "$A$") has the same elements as $A$, and multiplication corresponds to "AND", while addition corresponds to "XOR".

Recall also that Stone duality gives an equivalence between the opposite of the category of Boolean algebras and totally disconnected compact Hausdorff spaces. Under this equivalence, a Boolean algebra $A$ is sent to the space $Spec A$ of ultrafilters on $A$, and $A$ is recovered as the algebra of clopen subspaces of $Spec A$.

**Question:** Let $A$ be a Boolean ring. Let $M$ be an $A$-module. How can the data of $M$ be described in terms of the Boolean algebra $A$, or better yet in terms of the topological space $Spec A$?

One thing to say is that $M$ is naturally an $\mathbb F_2$-vector space, and the $A$-module structure on $M$ corresponds to a representation of $A$ as a sublattice of the lattice of $\mathbb F_2$-subspaces of $M$. This is nice as far as it goes, but I'd really like a description which doesn't mention vector spaces at all, just like the usual definitions of Boolean algebras or totally disconnected compact Hausdorff spaces don't mention rings at all. For instance, it would be nice if this could be described as some kind of representation of the Boolean algebra $A$ on the powerset lattice of a set or something like that.

**One possible direction:** If $M$ is an $A$-module, then there is a natural preorder on $M$ where $m \leq m'$ iff there is $a \in A$ such that $m = am'$. The set $Spec M$ of ultrafilters on this preorder carries a natural topology with subbasis given by the sets $\hat m = \{p \in Spec M \mid m \in p\}$, for $m \in M$. There is a natural continuous map $Spec M \to Spec A$ given by $p \mapsto \{a \in A \mid \exists m \in p (am = m)\}$. This yields a faithful functor $Spec : Mod_A^{op} \to Top_{/Spec A}$. I wonder if there some additional structure / properties on $Spec M$ which can turn this functor in an equivalence?

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