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A module M is quasi-injective iff it is injective over its own submodules, that is, given a submodule A of M and a homomorphism f of A to M, f lifts to an endomorphism of M. Equivalently, M is quasi-injective iff it is a fully invariant submodule of some injective module; iff it is a fully invariant submodule of its injective envelope.

A ring R is a left QI ring if and only if every quasi-injective left R-module is injective. Only a few non-trivial examples of QI rings are known, and in particular there are no known examples of rings which are QI on one side only. Boyle's Conjecture is almost 50 years old: Every left QI ring is left hereditary.

It turns out that the model theory of modules has a lot to say about the structure of and relations between the indecomposable injective left modules over a left QI ring. Perhaps these techniques will allow us to resolve some of the long-outstanding issues surrounding QI rings.

I will report on my recent and ongoing work with Ivo Herzog on these problems.