Additive unit structure of endomorphism rings and invariance of modules

Abstract

We use the type theory for rings of operators due to Kaplansky to describe the structure of modules that are invariant under automorphisms of their injective envelopes. Also, we highlight the importance of Boolean rings in the study of such modules. As a consequence of this approach, we are able to further the study initiated by Dickson and Fuller regarding when a module invariant under automorphisms of its injective envelope is invariant under any endomorphism of it. In particular, we find conditions for several classes of noetherian rings which ensure that modules invariant under automorphisms of their injective envelopes are quasi-injective. In the case of a commutative noetherian ring, we show that any automorphism-invariant module is quasi-injective. We also provide multiple examples that show that our conditions are the best possible, in the sense that if we relax them further then there exist automorphism-invariant modules which are not quasi-injective. We finish this paper by dualizing our results to the automorphism-coinvariant case.