Invariance of Modules under Automorphisms of their Envelopes and Covers

"The study of modules which are invariant under the action of certain subsets of the endomorphism ring of their injective envelope can be drawn back to the pioneering work of Johnson and Wong in which they characterized quasi-injective modules as those modules which are invariant under any endomorphism of their injective envelope. Later, Dickson and Fuller studied modules which are invariant under the group of all automorphisms of their injective envelope and proved that any indecomposable automorphism-invariant module over an F-algebra A is quasi-injective provided that F is a field with more than two elements. But after that this topic remained in dormant stage for some time until Lee and Zhou picked it up again in their paper where they called such modules auto-invariant modules. But the major breakthrough on this topic came from two papers that appeared a few months later: one of them was a paper of Er, Singh and Srivastava where they proved that the automorphism-invariant modules are precisely the pseudo-injective modules studied earlier by Teply, Jain, Clark, Huynh and others. The other one was a paper by Guil Asensio, and Srivastava where they proved that automorphism-invariant modules satisfy the exchange property and also they provide a new class of clean modules. Soon after this Guil Asensio and Srivastava extended the result of Dickson and Fuller by proving that if A is an algebra over a field F with more than two elements, then a module over A is automorphism-invariant if and only if it is quasi-injective. In 2015, in a paper published in the Israel Journal of Mathematics, Guil Asensio, Tutuncu and Srivastava laid down the foundation of general theory of modules invariant under automorphisms (resp. endomorphisms) of envelopes and covers. In this general theory of modules invariant under automorphisms (resp. endomorphisms) of envelopes and covers, we have obtained many interesting properties of such modules and found examples of some important classes of modules. When this theory is applied tosome particular situations, then we obtain results that extend and simplify several results existing in the literature. For example, as a consequence of these general results, one obtains that modules invariant under automorphisms of their injective (resp., pure-injective) envelopes satisfy the full exchange property. These results extend well-known results of Warfield, Fuchs, Huisgen-Zimmermann and Zimmermann. Most importantly, this study yields us a new tool and new perspective to look at generalizations of injective, pure-injective or at-cotorsion modules. Until now most of the generalizations of injective modules were focussed on relaxing conditions on lifting of homomorphisms but this theory has opened up a whole new direction in the study of moduletheory"--