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We use the type theory for rings of operators due to Kaplansky to describethe structure of modules that are invariant under automorphisms of theirinjective envelopes. Also, we highlight the importance of Boolean rings in thestudy of such modules. As a consequence of this approach, we are able tofurther the study initiated by Dickson and Fuller regarding when a moduleinvariant under automorphisms of its injective envelope is invariant under anyendomorphism of it. In particular, we find conditions for several classes ofnoetherian rings which ensure that modules invariant under automorphisms oftheir injective envelopes are quasi-injective. In the case of a commutativenoetherian ring, we show that any automorphism-invariant module isquasi-injective. We also provide multiple examples that show that ourconditions are the best possible, in the sense that if we relax them furtherthen there exist automorphism-invariant modules which are not quasi-injective.We finish this paper by dualizing our results to the automorphism-coinvariantcase.

Additive unit structure of endomorphism rings and invariance of modules