Countably-categorical Boolean rings with distinguished ideals

We describe and classify countable Boolean rings (which may or may not have a multiplicative identity) with finitely many distinguished ideals whose elementary theory is countably categorical. This extends the description by Macintyre and Rosenstein and subsequent authors of countably categorical Boolean algebras with finitely many distinguished ideals. Following Pierce, we take a topological approach using the language of PO systems (partially ordered sets with a distinguished subset) and topological Boolean algebras. We provide two different classifications via invariants that uniquely determine the isomorphism type: one using finite PO systems and the other using finite posets. We discuss how our findings link with previous results, but the paper is otherwise self-contained.